Decentralized risk-sharing:de…nitions, properties and characterizations
Pooling risks without a central insurer

Jan Dhaene
KU Leuven, Belgium

Chicago, Illinois, July 8, 2024

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Main references1
I Convex order and comonotonic conditional mean risk

sharing. Denuit M. & Dhaene J. (2012). Insurance: Mathematics
and Economics, 51, 249-256.
I Risk-sharing rules and their properties, with applications

to peer-to-peer insurance. Denuit M., Dhaene J. & Robert
C.Y. (2022). Journal of Risk and Insurance, 89(3), 615-667.
I Comonotonicity and Pareto optimality, with application

to collaborative insurance. Denuit M., Dhaene J., Ghossoub
M. & Robert C.Y. (2023). LIDAM Discussion Paper ISBA 2023/05.
I An axiomatic theory for quantile-based risk sharing.

Dhaene J., Robert C.Y, Cheung K.C., Denuit M. (2023). LIDAM
Discussion Paper ISBA 2023/28.
1 Hereafter, we will refer to these papers as D,D (2012), D,D,R (2022),

D,D,G,R (2023) and D,R,C,D (2023).
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AgendaI 1. Introduction.
I 2. Risk-sharing and risk-sharing rules.
I 3. Examples of risk-sharing rules.
I 4. Internal risk-sharing rules.
I 5. Aggregate risk-sharing rules.
I 6. Properties of risk-sharing rules.
I 7. Axiomatic characterization of the CMRS rule.
I 8. Further topics (not covered in this talk).
I References.

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1. Introduction
Risk pooling
I At time 0 (now), consider a pool of agents, each with arandom future loss at time 1.
I Centralized risk pooling:
I

Refers to risk transfer mechanisms where the individual losses
faced by the members of the pool are passed to a central
insurer.

I

Each insured in the insurance portfolio is compensatedex-post from the insurer for his loss.

I

In return, the insurer charges an ex-ante premium to each
insured and sets up a solvency capital at initiation.

I

The premiums follow from an appropriate premium principle.

I

Premiums and solvency capital are set such that theprobability that their sum at time 1 exceeds the aggregate lossof the insurance portfolio is su¢ ciently high.
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1. Introduction
Risk pooling
I At time 0 (now), consider a group of agents, each with arandom future loss at time 1.
I Decentralized risk pooling:
I

Refers to risk-sharing (RS) mechanisms under which theparticipants in the pool share their risks among each other.

I

Each participant in the risk-sharing pool is compensatedex-post from the pool for his loss.

I

In return, each participant pays an ex-post contribution to
the pool.

I

These contributions follow from a risk-sharing rule, which is
chosen such that the sum of all individual contributions to thepool is equal to the aggregate loss of the pool.

I

A decentralized approach is self-…nancing, implying that it
does not require setting up a solvency capital.
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1. Introduction
Risk pooling
I Risk-sharing pools of i.i.d. losses:
I

A natural choice for i.i.d.2 losses is the uniform risk-sharing
rule, where each participant contributes ex-post an equal partof the aggregate loss.

I Risk-sharing pools of losses which are not i.i.d.:I

In this case, the choice of an appropriate and simple RS rule isoften not straightforward.

I

A possible choice is the conditional mean RS (CMRS) rule3 ,where each participant contributes ex-post the conditional
expectation of his loss brought to the pool, given the observed
aggregrate loss covered by the pool.

2 i.i.d. = independent and identically distributed.
3 See D,D (2012).
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1. Introduction
Decentralized insurance
I Decentralized insurance: ’insurance’based on adecentralized risk pooling strategy:I

A group of participants bring their losses together in a RS pool.
The pool covers the losses of its participants,
I against a contribution to be paid ex post by each member.
I The r.v.’s describing the contributions are agreed upon ex ante,
I and chosen such that the self-…nancing condition is satis…ed.
I The role of the ’insurer’is restricted to ’administrator’.
I

I Synonyms: collaborative insurance, crowdsurance, or

peer-to-peer (P2P) insurance.
I P2P insurance is sometimes restricted to risk sharing withtransfers directly between participants rather than relying onan administrator who …rst collects the contributions and thenredistributes them among participants.
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1. Introduction
Decentralized insurance

I The group of participants who form the pool can consist of:I

friends, family members, a¢ nity groups, patients su¤eringfrom a same disease, etc.

Ilawyers, farmers, physicians, etc., who form a risk pool toprotect themselves against professional risks.

I

agents exposed to natural catastrophes or major industrial risks(induced by nuclear plants e.g.).

I Decentralized insurance revives early forms of mutual

insurance (where the contributions of the many are used tocover the losses of the few ).

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1. Introduction
Decentralized insurance
I Risk-sharing mechanisms have been studied for decades in

the actuarial literature, starting with the work of Karl Borchon equilibrium in reinsurance markets in the 1960’s.4
I The recent interest in a sharing economy, collaborative

consumption and decentralized …nance, together withrecent advances in technology have made decentralized
insurance a viable candidate to disrupt the traditional
insurance sector.
I In response to this recent evolution, actuarial researchers

have shown a renewed interest in the math supporting
decentralized insurance5 .
4 See e.g. Borch, K. (1962). Equilibrium in a reinsurance market.

Econometrica 3, 424-444.
5 See e.g. the website of FADeRiS, a series of workshops on the Foundations
and Applications of Decentralized Risk-Sharing, since 2023.
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1. Introduction
Conventions and notations
I Time 0 is ’now’.
I All r.v.’s are real-valued and de…ned on the atomlessprobability space (Ω, F , P).
I ω 2 Ω denotes the realized state-of-the-world at time 1.
I A random vector is denoted by a bold upper-case letter with

subscript indicating its dimension, e.g.
Xn = (X1 , X2 , . . . , Xn ) ,I while its realization (observed at time 1) is denoted by

Xn (ω ) = (X1 (ω ) , X2 (ω ) , . . . , Xn (ω ))10 / 88

2. Risk-sharing and risk-sharing rules
Agents and their losses

I Let χ be an appropriate (su¢ ciently rich) set of r.v.’s on(Ω, F , P), representing possible random losses at time 1.6
I Consider n economic agents, numbered i = 1, 2, . . . , n.
I Each agent i faces a loss Xi 2 χ at the end of the observation

period [0, 1].

I Without pooling, each agent bears his own loss:

At time 1, agent i su¤ers loss Xi (ω ).

6 If not explicitely stated di¤erently, we assume that χ = L1 .
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2. Risk-sharing and risk-sharing rules
Pools of losses

I The joint cdf of the loss vector Xn is denoted by FX .
n

I The marginal cdf’s of the individual losses are denoted by

FX 1 , FX 2 , . . . , FX n , respectively.
I The aggregate loss faced by the n agents with loss vector Xn
is denoted by SXn = ∑ni=1 Xi .
I Hereafter, we will often call Xn the pool (of losses) and eachagent a participant.
I In case no confusion about n is possible, we will write X

instead of Xn and SX (or S) instead of SXn .

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2. Risk-sharing and risk-sharing rules
Reallocations

I De…nition: For any pool X 2 χn with aggregate loss SX , theset AX (χn ) is de…ned by:n

AX ( χ ) =(n( Y1 , Y2 , . . . , Yn ) 2 χ j ∑ Yi = SX
n

i =1

)

I The elements of AX ( χn ) are called the n-dimensional
reallocations of X in χn .
I In case no confusion is possible, we will write AX instead of
AX ( χn ).

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2. Risk-sharing and risk-sharing rules
Risk-sharing
I De…nition: Risk-sharing in a pool X 2 χn is a two-stage

process.
I

Ex-ante step (at time 0):
The losses Xi in the pool X are reallocated by transforming X
into another random vector C [X] 2 AX :C [ X ] = C1 [ X ] , C2 [ X ] , . . . , Cn [ X ]I

Ex-post step (at time 1):
I
I

Each participant i receives from the pool his loss Xi .
In return, he contributes to the pool his reallocated loss
C i [X].

I Self-…nancing condition:

As C [X] 2 AX , risk-sharing is fully reallocating the loss
vector X:
n

n

i =1

i =1

∑ C i [ X ] = ∑ Xi
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2. Risk-sharing and risk-sharing rules
Risk-sharing rules

I Consider a group of n agents (participants) and a set of loss
vectors χn .
I De…nition:

A risk-sharing rule for the group of nparticipants is a mapping C : χn ! χn which transforms any
loss vector X into a contribution vector C [X]:X 2 χn ! C [X] 2 AX ( χn )I Terminology:
I

Consider the n participants and suppose their loss vector is X.

I

Xi is the loss of participant i (paid by the pool).

I

Ci [X] is the contribution of participant i (paid to the pool).

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2. Risk-sharing and risk-sharing rules
Risk-sharing rules
I Risk-sharing rules can be complex mappings.
I The contribution vector C [X] may depend on a priori info

(i.e. info known at time 0) concerning:Ilosses: expectations of the Xi and/or cdf of X.

I

participants: their ages, their initial wealth, theirplace-of-residence, their ID number, their utility functions.

I

other: current state-of-the-economy.

I The contribution vector C [X] may also depend on

a posteriori info (i.e. time-1 observable r.v.’s) concerning:Ilosses: the losses Xi and/or the aggregate claims S.

I

participants: mileage increase of their cars, telematics info
about their driving behavior.

Iother: in‡ation index at time 1.
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2. Risk-sharing and risk-sharing rules
Risk-sharing rules

Example 2.1:
I Consider two agents with age vector x = (x1 , x2 ) and loss
vector in χ2 .
I Consider the RS rule C : χ2 ! χ2 , where for any pool X 2 χ2the contribution vector is given by
C [X] =

x2
x1
SX ,
SX
x1 + x2
x1 + x2

I Interpretation:

The older participant contributes a higher proportion of theaggregate claims.

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2. Risk-sharing and risk-sharing rules
Risk-sharing rules

Example 2.2:
I Consider a group of n car drivers with loss vector in χndescribing their third party liability claims.
I Let us denote the increases in mileage of the cars of theseagents by the random vector Z = (Z1 , Z2 , . . . , Zn ).
I For these car drivers, consider the RS rule C : χn ! χn ,where for any pool X 2 χn , one has

C [X] =Zn
Z1
SX , . . . ,SXZ1 + . . . + Zn
Z1 + . . . + Zn

I Interpretation:

Participants pay a proportion of the aggregate claims basedon the increase in mileage of the car they are driving.
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2. Risk-sharing and risk-sharing rules
Risk-sharing rules

Example 2.3:
I Consider a group of n participants:
I

These participants can be divided in m age categories, denotedby 1, 2, . . . , m.

I

The age category of participant j is denoted by aj .

I Consider the RS rule C : χn ! χn , where for any poolX 2 χn , the contribution vector follows frommCi [ X ] = ∑k =1∑nj=1 Xj 1aj =k∑nj=1 1aj =k

1ai =k ,i = 1, 2, . . . , n

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2. Risk-sharing and risk-sharing rules
Risk-sharing rules

Example 2.3 (cont’d):
I The RS rule C is de…ned bymCi [ X ] = ∑k =1∑nj=1 Xj 1aj =k∑nj=1 1aj =k1a i = k

I Interpretation: Losses are uniformly shared within each age

category.

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2. Risk-sharing and risk-sharing rules
Combining risk retention, risk-sharing and risk transfer
I In order to reduce his contribution to the pool, a participant

with initial loss Yi may:I

retain part of Yi
transfer part of Yi to an insurer,
I share the remaining part of Y with the members of a pool.
i
I

I In this presentation:I

Xi denotes the loss that i brings to the the pool, after eventual
retention and transfer of risk.
I C [X] denotes the contribution of i to the pool.
i
I Example:
I

Suppose that participant i retains the lower layer [0, li ) of
initial loss Yi
0,
I and transfers the upper layer [u , ∞ ) of Y , with u > l to aniiiiinsurer.
I The loss shared in the pool is thenXi = (Yi

li )+

(Yi

ui ) +
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3. Examples of risk-sharing rules
The stand-alone risk-sharing ruleI De…nition: C : χn ! χn is the stand-alone RS rule if

C [X] = X,

for any X 2 χn

I Interpretation: This RS rule corresponds to the case where

the agents decide not to pool their risks.
I The pool just acts as a register to collect data about individual

losses, without re-allocating them among participants.

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3. Examples of risk-sharing rules
The uniform risk-sharing ruleI De…nition: The uniform RS rule Cuni : χn ! χn is de…nedbyCiuni [X] =SX,ni = 1, 2, . . . , n,

for any X 2 χn .
I Interpretation: This RS rule equally distributes the aggregate

loss SX over all participants.
I It is the most simple, non-trivial and well-known RS rule.

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3. Examples of risk-sharing rules
The mean-proportional risk-sharing ruleI De…nition: The mean-proportional RS rule
Cprop : χn ! χn is de…ned by
prop

Ci

[X] =

for any X 2 χn

E [ Xi ]SX ,E [ SX ]i = 1, 2, . . . , n.

n

L1+ .

I Interpretation: The participants agree to take a …xed

percentage of the total loss SX , in accordance with the
expected values of the losses they bring to the pool.
I Proportional rules have been widely used in a risk-sharing

context in practice.

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3. Examples of risk-sharing rules
The order statistics risk-sharing rule
I The i-th order statistic X(i ) of a random vector

X = (X1 , X2 , . . . , Xn ) is the i-th smallest value in X. Hence,
X(1 ) X(2 ) . . . X(n ) .
I De…nition: The order statistics RS rule Cord : χn ! χn isde…ned by

Cord [X] = X(1 ) , X(2 ) , . . . , X(n )for any X 2 χn .
I Interpretation: For participants which are ordered inascending risk-bearing capacity (e.g. age), a higher
risk-bearing capacity leads to a higher contribution.

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3. Examples of risk-sharing rules
The conditional mean risk-sharing ruleI De…nition: The conditional mean RS rule Ccm : χn ! χnis de…ned by

Cicm [X] = E [Xi j SX ] ,i = 1, 2, . . . , n,n

for any X 2 χn = L1+ .
I Interpretation: Each participant contributes the expectedvalue of the loss he brings to the pool, given the aggregate
loss experienced by the pool.
I The CMRS rule was introduced in the actuarial literature in
D,D (2012)7 . It has many nice properties, see D,D,R (2022).

Its axiomatic characterization is given in Jiao, Kou, Liu &Wang (2022).
7 An early reference is Landsberger & Meilijson (1994).
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3. Examples of risk-sharing rules
The conditional mean risk-sharing rule
I A reshu- e of the loss vector X = (X1 , X2 , . . . , Xn ) is aπrandom vector X de…ned by

Xπ = Xπ (1 ) , Xπ (2 ) , . . . , Xπ (n ) ,where π = (π (1), π (2), . . . , π (n )) is a permutation of
f1, . . . , ng.
I Interpretation: X and Xπ are composed of the same

individual losses, but only their places are interchanged: After
reshu- ing, Xπ (i ) is the new loss attributed to participant i.
I De…nition:8 The loss vector X is exchangeable in case fordany reshu- e π of X, one has that Xπ = X.
d
8 Xπ =
X is a notation for ’Xπ and X have the same cdf’.
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3. Examples of risk-sharing rules
The conditional mean risk-sharing rule

I The individual losses of an exchangeable loss vector are i.d.9 ,but not necessary mutually independent.
I Property:

In case the loss vector X 2 χn is exchangeable, thenCcm [X] = E [X j SX ] =

SX
SX SX
,
,...,
n n
n

I Interpretation:

For exchangeable loss vectors, the conditional mean RS rule
reduces to the uniform RS.

9 The notation ’i.d.’ stands for ’identically distributed’.
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4. Internal risk-sharing rules
De…nition

I De…nition:

C : χn ! χn is a internal RS rule if for any loss vector
X 2 χn , one has that C [X] is X-measurable.
I Interpretation:

C is internal in the sense that the randomness of any
contribution vector C [X] is solely due to the randomness of
the loss vector X.

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4. Internal risk-sharing rules
De…nition
I Other characterizations:C : χn ! χn is an internal RS rule if and only if any of thefollowing equivalent conditions holds:(1) For any X 2 χn , there exists a function hint : Rn ! Rn suchthat
C [X] = hint (X)(2) For any X 2 χn , one has that
C [X] = E [C [X] j X]
I Remark:

The function hint may be pool-speci…c, in the sense that it
may be di¤erent for di¤erent loss vectors.
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4. Internal risk-sharing rules
Examples of internal RS rules
I Stand-alone RS: internal,

Ci [ X ] = X
I Uniform RS: internal,

Ciuni [X] =SXnI Proportional RS: internal,
prop

Ci

[X] =

E [ Xi ]SX
E [ SX ]I Order statistics RS: internal,

Ciord [X] = X(i )I Conditional mean RS: internal,

Cicm [X] = E [Xi j SX ]31 / 88

4. Internal risk-sharing rules
Counterexample

Example 2.4:
I At time 1, we ‡ip a coin. The r.v. Z equals 0 if the coin lands

on heads and 1 if it lands on tails.
I Consider the RS rule C : χ2 ! χ2 , which is de…ned such thatfor any pool X = (X1 , X2 ),C1 [ X ] =X1 + X2 : Z = 00:Z =1C2 [ X ] =0:Z =0
X1 + X2 : Z = 1

and

I Observations:I

I

It is impossible to …nd for any pool X a function
hint : R2 ! R2 such that C [X] = hint (X).
This means that C is not an internal RS rule.
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4. Internal risk-sharing rules
Special classes of internal risk-sharing rules
I C is type 1 - internal if there exists a function h(1) such thatfor all X, one has that;
C [X] = h(1) (X; FX )I C is type 2 - internal if there exists a function h(2) such thatfor all X, one has that:
C [X] = h(2) (X; FX 1 , FX 2 , . . . , FX n )I C is type 3 - internal if there exists a function h(3) such thatfor all X, one has that:
C [X] = h(3) (X)33 / 88

5. Aggregate risk-sharing rules
De…nition

I De…nition:

RS rule C : χn ! χn is an aggregate RS rule if for any poolX 2 χn , one has that C [X] is SX -measurable.
I Interpretation:

C is aggregate in the sense that the randomness of any
contribution vector C [X] is solely due to the randomness ofaggregate claims SX .
I Remark:

An aggregate RS rule is anonymous, in the sense that it
doesn’t care about the ’origin’of the aggregate loss SX .

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5. Aggregate risk-sharing rules
De…nition
I Other characterizations:C : χn ! χn is an aggregate RS rule if and only if any of
the following equivalent conditions holds:(1) For any X 2 χn , there exists a function haggr : R ! Rn suchthat
C [X] = haggr (SX )(2) For any X 2 χn , one has that
C [X] = E [C [X] j SX ]I Remark:

The function haggr may be pool-speci…c, in the sense that it
may be di¤erent for di¤erent loss vectors.
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5. Aggregate risk-sharing rules
De…nition
I Property:

An aggregate RS rule is an internal RS rule.
I Proof:I
I

Let C : χn ! χn be an an aggregate RS rule and X 2 χn .
There exists a function haggr : R ! Rn such that
C [X] = haggr (SX )I

De…ne the function hint : Rn ! Rn such thathint (x) = haggr (x1 + . . . + xn )I

I

Then we …nd that

C [X] = hint (X)We can conclude that C is an internal RS rule.

5
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5. Aggregate risk-sharing rules
ExamplesI Stand-alone RS: not aggregate,

C [X] = X
I Uniform RS: aggregrate,

SXn
I Mean-proportional RS: aggregate,
Ciuni [X] =

prop

Ci

[X] =

E [ Xi ]SX
E [ SX ]I Order statistics RS: not aggregate,

Ciord [X] = X(i )I Conditional mean RS: aggregate,

Cicm [X] = E [Xi j SX ]37 / 88

5. Aggregate risk-sharing rules
Classi…cation of aggregate risk-sharing rules
I C is type 1 - aggregate if there exists a function h(1) suchthat for all X, one has that:
C [X] = h(1) (SX ; FX )I C is type 2 - aggregate if there exists a function h(2) suchthat for all X, one has that:
C [X] = h(2) (SX ; FX 1 , FX 2 , . . . , FX n )I C is type 3 - aggregate if there exists a function h(3) suchthat for all X, one has that:
C [X] = h(3) (SX )38 / 88

6. Properties of risk-sharing rules
Introduction
I Premium principles: have been studied in detail around the’eighthies’10 .
I Risk measures: During the the ’nineties’and beyond, risk

measures and their axiomatic characterizations have beenstudied extensively11 .
I Risk-sharing rules: Recently, the study of decentralized RS

has (re-)gained interest.
I

Hereafter, we present a non-exhaustive list of properties that
RS rules should/could satisfy12 .
I Some properties are inspired by properties of premium
principles or risk measures, other are tailored to RS rules.

10 Goovaerts, De Vijlder & Haezendonck (1984) is a key reference.
11 Artzner, Delbaen, Eber & Heath (1997) is a fundamental paper.
12 See D,D,R (2022).
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6. Properties of risk-sharing rules
Classes of properties
I Conservation properties: also hold for the stand-alone RS

rule: reshu- ing, normalization, translativity, positive
homogeneity, constancy, no rip-o¤, actuarial fairness, law
invariance.
I Improvement properties: guarantee that the RS rule

’improves’the situation compared to the stand-alone
situation: willingness-to-join, comonotonicity.
I Coalition properties: guarantee that participants are nota¤ected by coalitions set up between other participants:
coalition fairness, merging fairness.
I Speci…c pool properties: should hold (or not) for speci…cpools only: stand-alone property for comonotonic pools,
uniformity property for exchangeable pools.
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6.1. Conservation properties of risk-sharing rules
The ’reshu- ing’property
I De…nition:13 A RS rule C : χn ! χn satis…es the reshu- ing
property if for any X 2 χn and any of its reshu- es Xπ , one

has thatCi [ X π ] = C π ( i ) [ X ]for any i = 1, . . . , n

I Example:
I

Consider X = (X1 , X2 , X3 ) and Xπ = (X3 , X1 , X2 ).

I

As π (1) = 3, the reshu- ing property implies that
C1 [Xπ ] = C3 [X]

I Interpretation : If participants exchange their individual

losses, then their contributions are changed in the same way.
13 Jiao, Kou, Liu & Wang (2022), hereafter abbreviated as J,K,L,W (2022),

call this property symmetry.
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6.1. Conservation properties of risk-sharing rules
The ’reshu- ing’property

I Order statistics RS: does not satisfy reshu- ing,

C1ord [Xπ ] = C1ord X3 , X1 , X2 = min X1 , X2 , X3
while
ord
Cπord
X1 , X2 , X3 = max X1 , X2 , X3
( 1 ) [ X ] = C3

I Conditional means RS: satis…es reshu- ing,hiCicm [Xπ ] = E Xπ (i ) j S = Cπcm(i ) [ X ]42 / 88

6.1. Conservation properties of risk-sharing rules
The ’reshu- ing’property
I Theorem:

Consider the RS rule C : χn ! χn .
I C is the uniform RS rule if and only if it satis…es thefollowing axioms:

I(1) C is a type 3 - aggregate RS rule:
C [X] = h(3) (SX )(2) C satis…es reshu- ing:
C i [ Xπ ] = C π (i ) [ X ]I Proof:I
I) : Straightforward.
( For any reshu- e Xπ of X and any participant i, we have:(ax.1)(3)

C i [ X ] = hi

(3)(SX ) = hi(ax.1)(ax.2)( SX π ) = Ci [ X π ] = C π ( i ) [ X ]We can conclude that all Ci [X] are equal.

5

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6.1. Conservation properties of risk-sharing rules
The ’reshu- ing’property
I Proposition:

The axioms (1) and (2) are independent:
For any choice of a particular axiom, there always exists a RS
rule C : χn ! χn which satis…es that axiom but not the other.
I Proof:I

The RS rule C : χn ! χn , given by

X1
S
Xn
S
+ X,...,
+ X
2
2n
2
2n
satis…es axiom (2), but not axiom (1).
C [X] =I

For n 6= 2, the RS rule C : χn ! χn given by
C [X] =

SX
SX
SX
,
,...,
2 2(n 1)2(n 1)

satis…es axiom (1), but not axiom (2).

544 / 88

6.1. Conservation properties of risk-sharing rules
The ’normalization’property
I De…nition:14 A RS rule C : χn ! χn satis…es thenormalization property if for any X 2 χn , one has

Xi = 0 ) C i [ X ] = 0
holds for any i.
I Interpretation: A participant with a zero loss r.v. has a zero

contribution.
I Remark: If a RS rule satis…es the reshu- ing property, then asu¢ cient condition for the normalization property to hold isthat it holds for the last participant.
14 J,K,L,W (2022) call this property zero preserving. Normalization as de…ned

here is a weaker property than normalization de…ned in D,D,R (2022).
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6.1. Conservation properties of risk-sharing rules
The ’normalization’property

I Uniform RS: does not satisfy normalization,

C3uni X1 , X2 , 0 =X1 + X2
3

I Conditional means RS: satis…es normalization,I

The CMRS satis…es reshu- ing.

I

Furthermore,
Cncm X1 , X2 , . . . , Xn 1 , 0 = E [0 j S ] = 0

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6.1. Conservation properties of risk-sharing rules
The ’translativity’property
I Notation: ej is the n-dimensional unit vector with allcomponents equal to 0, except the j-th component whichequals 1.
I De…nition: A RS rule C : χn ! χn satis…es the translativity
property if for any X 2 χn and any participant j = 1, . . . , n,

one has thatCi [X + c ej ] = Ci [X] ,for all i 6= j and any c

0

I Interpretation: Increasing the loss of participant j by adeterministic amount c 0, leaves the contributions of theother participants unchanged.
I Consequence: In case the RS rule C satis…es the translativity

property, thenCj [X + c ej ] = Cj [X] + c
47 / 88

6.1. Conservation properties of risk-sharing rules
The ’translativity’property
I Remark: If the RS rule satis…es the reshu- ing property, thena su¢ cient condition for the translativity property to hold isthat it holds for the last participant.
I Uniform RS: does not satisfy translativity,

C1uni X1 , X2 , X3 + c =S +cS
6= = C1uni X1 , X2 , X3
3
3

I Conditional mean RS satis…es translativity,
I

The CMRS satis…es reshu- ing.

I

For any i 6= n, one has
Cicm X + c 1n = E[Xi j jS + c ] = E[Xi j S ] = Cicm X

48 / 88

6.1. Conservation properties of risk-sharing rules
The ’positive homogeneity’property

I De…nition:

A RS rule C : χn ! χn satis…es the positive homogeneity
property15 if for any X 2 χn , one has thatCi [ c X ] = cholds for any i and any c

Ci [ X ]0.

I Interpretation: Multiplying all losses by c

0 implies that

all contributions change in the same way.

15 c X stands for (cX1 , c

X2 , . . . , cXn )49 / 88

6.1. Conservation properties of risk-sharing rules
The ’positive homogeneity’property

I Mean-proportional RS is positive homogeneous,
prop

Ci

E[c Xi ](c S )
E[c S ]
E [ Xi ]prop=c
S = c Ci[X]
E[S ][c X] =I Conditional mean RS is positive homogeneous,Cicm [c X] = E[cXi j cS ] = c E [ Xi j S ] = c

Cicm [X]50 / 88

6.1. Conservation properties of risk-sharing rules
The ’constancy’property
I De…nition:16 A RS rule C : χn ! χn satis…es the constancy
property if for any X 2 χn , one has

Xj = c ) C j [ X ] = cholds for any j and any c

0.

I Interpretation: In case a participant’s loss is a non-negative

constant, then his contribution is equal to that constant.
I Remark: If the RS rule satis…es the reshu- ing property, thena su¢ cient condition for the constancy property to hold isthat it holds for the last participant.
16 Constancy as de…ned here is a weaker property than the constancy de…ned

in D,D,R (2022). Constancy is called constant preserving in J,K,L,W (2022).
51 / 88

6.1. Conservation properties of risk-sharing rules
The ’constancy’property

I Uniform RS: does not satisfy constancy,

C3uni X1 , X2 , c =X1 + X2 + c
3I Mean-proportional RS: does not satisfy no-ripo¤,
prop

C3

X1 , X2 , c =

c
S
E[S ]

I Conditional mean RS: satis…es constancy,

Cncm X1 , X2 , . . . , Xn 1 , c = E[c j S ] = c

52 / 88

6.1. Conservation properties of risk-sharing rules
The ’constancy’property

I Proposition:

If a RS rule satis…es translativity and normalization, then it
also satis…es constancy.
I Proof:I

The case where Xn = c:Cn X1 , X2 , . . . , Xn 1 , c = Cn X1 , X2 , . . . , Xn 1 , 0 + c
transl.

= Cn X1 , X2 , . . . , Xn 1 , 0 + c

normaliz.

=Ic

The case where Xj = c for j 6= n is proven similarly. 5

53 / 88

6.1. Conservation properties of risk-sharing rules
The ’no rip-o¤’property

I The ’largest value’of r.v. X :FX 1 (1) = inf fx 2 R j FX (x ) = 1g
I De…nition:17 A RS rule C : χn ! χn satis…es the no rip-o¤
property if for any X 2 χn , one has thatCi [ X ]FX i 1 (1)holds for any i = 1, 2, . . . , n.
I Interpretation: The participant’s contribution will neverexceed his ’worst-case loss’.

17 J,K,L,W (2022) call this property the ’risk fairness’axiom.
54 / 88

6.1. Conservation properties of risk-sharing rules
The ’no rip-o¤’property
I Uniform RS: does not satisfy no-ripo¤,18

C1uni U, 2U =3U
2 [0, 1.5]
2

I Proportional RS: does not satisfy no-ripo¤,
prop

C3

X1 , X2 , c =

c
S
E[S ]

I Conditional mean RS: satis…es no rip-o¤,

Cicm X = E[Xi j S ]E[FX i 1 (1) j S ] = FX i 1 (1) .

18 In this text, the notation U is used exclusively for a r.v. which is uniformly

distributed over [0, 1].
55 / 88

6.1. Conservation properties of risk-sharing rules
The ’actuarial fairness’property

I De…nition: A RS rule C : χn ! χn satis…es
actuarial fairness if for any X 2 χn , one has that

E Ci X= E [ Xi ]holds for any i = 1, 2, . . . , n.
I Interpretation: On average, participants do neither gain nor

lose from risk sharing, in the sense that their expected
contribution (by joining the pool) is equal to their expected
loss (when staying alone).

56 / 88

6.1. Conservation properties of risk-sharing rules
The ’actuarial fairness’property
I Uniform RS: does not satisfy actuarial fairness,
I
I

Consider the loss vector U, U, 4U .
Expected loss of …rst participant:
E[U ] =

I

1
2

Expected contribution of …rst participant:
E C1uni [U, U, 4U ] = E

U + U + 4U
=1
3

I Conditional mean RS: satis…es actuarial fairness,

E [Cicm [X]] = E E[Xi j S ] = E [Xi ]57 / 88

6.1. Conservation properties of risk-sharing rules
The ’actuarial fairness’property
I Proposition:

If a RS rule satis…es actuarial fairness and no rip-o¤, then it
also satis…es constancy.
I ProofI
I

Suppose Xi = c.
Actuarial fairness implies
E [Ci [X]] = E [Xi ] = c

INo rip-o¤ implies
Ci [X]I

FX 1 ( 1 ) = ci

We can conclude thatCi [ X ] = c

558 / 88

6.1. Conservation properties of risk-sharing rules
The ’law-invariance’property

I De…nition:

A RS rule C : χn ! χn satis…es the law-invariance property
in casedd
X = Y ) C [X] = C [Y ]holds for any X and Y 2 χn .
I Interpretation:

If pools are ’equal in distribution’, then also their contribution
vectors are ’equal in distribution’.

59 / 88

6.1. Conservation properties of risk-sharing rules
The ’law-invariance’propertyd
I Uniform RS: is law invariant: If X = Y, thenX1 + . . . + Xn
,...
n

d

=

Y1 + . . . + Yn
,...
n

d
I Conditional means RS: is law invariant: If X = Y, thend

(E [X1 j SX ] , . . . , E [Xn j SX ]) = (E [Y1 j SY ] , . . . , E [Yn j SY ])60 / 88

6.2. Improvement properties of risk-sharing rules
Classes of properties
I Conservation properties: also hold for the stand-alone RS

rule: reshu- ing, normalization, translativity, positive
homogeneity, constancy, no rip-o¤, actuarial fairness, law
invariance.
I Improvement properties: guarantee that the RS rule

’improves’the situation compared to the stand-alone
situation: willingness-to-join, comonotonicity.
I Coalition properties: guarantee that participants are nota¤ected by coalitions between other participants coalition
fairness, merging fairness.
I Speci…c pool properties: should hold (or not) for speci…cpools only: stand-alone property for comonotonic pools,
uniformity property for exchangeable pools.
61 / 88

6.2. Improvement properties of risk-sharing rules
Increasing convex order
I Consider the r.v.’s X and Y .
I De…nition:

X is smaller than Y in stop-loss order19 if for any
non-decreasing concave function u : R ! R and any real w ,one has thatE[u (wX )]E[u (wY )]I Interpretation in expected utility theory:

Risk-averse decision makers prefer loss X over loss Y .
I Notation:

X

sl Y

19 ’Stop-loss order’is also called ’increasing convex order’.
62 / 88

6.2. Improvement properties of risk-sharing rules
The ’willingness-to-join’property

I De…nition:

A RS rule C : χn ! χn satis…es the willingness-to-joinproperty20 if for any X 2 χn , one has thatCi [ X ]sl Xiholds for any i = 1, 2, . . . , n.
I Interpretation: Any risk-averse participant i will prefer thecontribution Ci [X] over his initial loss Xi .

20 In J,K,L,W (2022) this property is called the universal improvement

property.
63 / 88

6.2. Improvement properties of risk-sharing rules
The ’willingness-to-join’property
I Proposition: If a RS rule satis…es willingness-to-join, then it

also satis…es actuarial fairness, no rip-o¤ and constancy.
I Proof:I

For any i, willingness-to-join implies
E [Ci (X)]E [Xi ]

The full allocation condition states that
n

n

i =1

i =1

∑ E [Ci [X]] = ∑ E [Xi ]We can conclude that C is actuarially fair:
E [Ci [X]] = E [Xi ]I

Willingness-to-join implies no rip-o¤:
Ci [X]

I

FC 1[X] (1)i

FX 1 ( 1 )iActuarial fairness and no rip-o¤ imply constancy.

564 / 88

6.2. Improvement properties of risk-sharing rules
The ’willingness-to-join’property

I Uniform RS: does not satisfy willingness-to-join, since it

is not an actuarially fair RS rule.
I Proportional RS: does not satisfy willingness-to-join,since it does not satisfy the no rip-o¤ property.
I Conditional mean RS: satis…es willingness-to-join:

E [ Xi j S ]sl Xi

65 / 88

6.2. Improvement properties of risk-sharing rules
Comonotonicity

I Consider the pool X = (X1 , . . . , Xn ) with aggregate loss SX .
I De…nition: X is comonotonic if there exist non-decreasing

functions gi : R ! R such thatX = g1 (SX ), . . . , gn (SX )(1)

I Interpretation: X is comonotonic if the increase of one of theindividual losses implies an increase of all the other individual
losses.

66 / 88

6.2. Improvement properties of risk-sharing rules
The ’comonotonicity’property
I De…nition: A RS rule C : χn ! χn is comonotonic if for any
X 2 χn , one has that C [X] is a comonotonic random vector.
I Equivalent de…nition: A RS rule C is comonotonic if for
any X 2 χn , there exists a function hcom : R ! Rn such that

C [X] = hcom (SX ) = h1com (SX ), . . . , hncom (SX )where all functions hicom are non-decreasing.
I Interpretation:

Comonotonicity of a RS rule ensures that any participant has
an interest in keeping his loss as small as possible21 .
21 For this reason, Carlier & Dana (2003) call this property the no-sabotage

condition.
67 / 88

6.2. Improvement properties of risk-sharing rules
The ’comonotonicity’property
I Remark:I

C is comonotonic ) C is aggregate ) C is internal.

I

C is internal ; C is aggregate ; C is comonotonic.

I Uniform RS: comonotonic. Indeed, for any X, one has thatuni

C[X] is comonotonic:

Cuni [X] =

SX
SX
,...,
n
n

I Conditional mean RS: aggregate but not comonotonic

since the contribution vector
Ccm [X] = E[X1 j SX ], . . . , E[Xn j SX ]is not necessarily comonotonic.
68 / 88

6.3. Coalition properties of risk-sharing rules
Classes of properties
I Conservation properties: also hold for the stand-alone RS

rule: reshu- ing, normalization, translativity, positive
homogeneity, constancy, no rip-o¤, actuarial fairness, law
invariance.
I Improvement properties: guarantee that the RS rule

’improves’the situation compared to the stand-alone
situation: willingness-to-join, comonotonicity.
I Coalition properties: guarantee that participants are nota¤ected by coalitions between other participants: coalition
fairness, merging fairness.
I Speci…c pool properties: should hold (or not) for speci…cpools only: stand-alone property for comonotonic pools,
uniformity property for exchangeable pools.
69 / 88

6.3. Coalition properties of risk-sharing rules
The ’coalition fairness’property
I Suppose that the participants in the pool

X1 , X2 , X3 agree
to use the RS rule C to determine their contributions.

I Next, suppose that participants 2 and 3 decide to form a

coalition, under which they equally share their joint loss
( X2 + X3 ) .
I This coalition transforms the pool

X1 ,

X1 , X2 , X3 into

X2 + X3 X2 + X3,
2
2

I It seems reasonable to require that the contribution ofparticipant 1, who is excluded from this coalition, remains
unchanged.
I RS rules satisfying this property will be said to satisfy thecoalition fairness property.
70 / 88

6.3. Coalition properties of risk-sharing rules
The ’coalition fairness’property

I Recall:A reallocation Y of a pool X 2 χn is an element of thefollowing set:
)
(
n

AX =

( Y1 , Y2 , . . . , Yn ) 2 χ n j ∑ Yi = SXi

I De…nition:
I
I

Consider a group of n participants and a pool X 2 χn .
A coalition ’excluding participant i’is a reallocation
Y 2 AX of X, with Yi = Xi .

71 / 88

6.3. Coalition properties of risk-sharing rules
The ’coalition fairness’property
I De…nition:

A RS rule C : χn ! χn satis…es the coalition fairness
property if for any X 2 χn and any Y 2 AX , one has that
Yi = Xi ) Ci [Y ] = Ci [X]holds for any i.
I Interpretation:I

The contribution of a participant in a pool is not a¤ected bycoalitions between other participants in this pool.

I

A coalition between a group of participants does not change
the joint contributions of this group.

I Remark: If a RS rule satis…es reshu- ing, then a su¢ cientcondition for coalition fairness to hold is that it holds for thelast participant.
72 / 88

6.3. Coalition properties of risk-sharing rules
The ’coalition fairness’property

I Consider the pools X and Y in AX , with Yi = Xi .
I Uniform RS: is coalition fair. Indeed,Ciuni [Y ] =SYS= X = Ciuni [X]nn

I Mean-proportional RS: is coalition fair. Indeed,

Ciprop [Y ] =E [ Yi ]E [ Xi ]SY =SX = Ciprop [X]
E [ SY ]E [ SX ]73 / 88

6.3. Coalition properties of risk-sharing rules
The ’coalition fairness’property
I Consider the pools X and Y in AX , with Yi = Xi .
I Conditional Mean RS: is coalition fair. Indeed,

Cicm [Y ] = E [Yi j SY ] = E [Xi j SX ] = Cicm [X]
I Order statistics RS: is not coalition fair. Indeed, consider

X1 , X2 , X3 and X1 , X2 + X3 , 0
One has that
C1ord X1 , X2 , X3 = min fX1 , X2 , X3 g
while
C1ord X1 , X2 + X3 , 0 = min fX1 , X2 + X3 , 0g
74 / 88

6.3. Coalition properties of risk-sharing rules
The ’coalition fairness’property

I Proposition:

A RS rule C : χn ! χn satis…es coalition fairness, for any X 2 χn and any participant i, one has thatCi [X] = Ci Xi ei + (SXXi ) ej

for any j 6= i

I Proposition:

If a RS rule satis…es normalization and coalition fairness,then it also satis…es reshu- ing.

75 / 88

6.4. Speci…c pool properties of risk-sharing rules
Classes of properties
I Conservation properties: also hold for the stand-alone RS

rule: reshu- ing, normalization, translativity, positive
homogeneity, constancy, no rip-o¤, actuarial fairness, law
invariance.
I Improvement properties: guarantee that the RS rule

’improves’the situation compared to the stand-alone
situation: willingness-to-join, comonotonicity.
I Coalition properties: guarantee that participants are nota¤ected by coalitions between other participants: coalition
fairness, merging fairness.
I Speci…c pool properties: should hold (or not) for speci…cpools only: stand-alone property for comonotonic pools,
uniformity property for exchangeable pools.
76 / 88

6.4. Speci…c pool properties of risk-sharing rules
The ’stand-alone for comonotonic pools’property

I De…nition: A RS rule C : χn ! χn satis…es thestand-alone property for comonotonic pools if for any
comonotonic pool X 2 χn , one has thatC X =XI Interpretation: In a comonotonic pool, no diversi…cation

bene…t arises from risk-sharing. Therefore, it may bereasonable to require that in such a pool each participant
remains with his own risk.

77 / 88

6.4. Speci…c pool properties of risk-sharing rules
The ’stand-alone for comonotonic pools’property

I Uniform RS: does not satisfy stand-alone forcomonotonic pools.
Indeed, for the comonotonic pool X = (U, 2U, U ), one has
C1uni U, 2U, U =4
U 6= U
3

I Conditional mean RS: satis…es stand-alone forcomonotonic pools.
Indeed, for the comonotonic pool X = g1 (SX ), . . . , gn (SX ) ,with all gi non-decreasing functions, one has
Cicm X = E [Xi j SX ] = E [gi (SX ) j SX ] = gi (SX ) = Xi

78 / 88

6.4. Speci…c pool properties of risk-sharing rules
The ’uniformity for exchangeable pools’property

I De…nition:

C : χn ! χn satis…es uniformity for exchangeable pools if
for any exchangeable pool X 2 χn , one has:
C X =

SX SX
SX
,
,...,
n n
n

I Interpretation:
I

The exchangeable pool X is homogeneous in the sense thatthe joint cdf of X is not changed by reshu- ing.

I

In this case, the uniform RS rule seems to be appropriate.

79 / 88

6.4. Speci…c pool properties of risk-sharing rules
The ’uniformity for exchangeable pools’property

I Order statistics RS: does not satisfy uniformity forexchangeable pools.
Indeed, for any exchangeable pool X, one has that
C1ord X = min fX1 , . . . , , Xn gwhich is not necessarily equal to SnX .
I Conditional mean RS: satis…es uniformity forexchangeable pools.
Indeed, for any exchangeable pool X and any i one has that
Cicm X = E [Xi j SX ] =SXn80 / 88

7. Axiomatic characterization of the CMRS rule
I Consider the CMRS rule Ccm :

n

L1+

n

! L1+ , with

Ccm [X] = E [X j SX ]I We have proven that Ccm satis…es the following properties:I

Ccm is an aggregate RS rule:Ccm [X] is SX

I

Ccm satis…es no rip-o¤: For any i,
Cicm [X]I

measurable

C

cm

FX 1 ( 1 )isatis…es actuarial fairness: For any i,
E [Cicm [X]] = E [Xi ]I

Ccm satis…es coalition fairness: For any Y 2 AX

L1+

n

,

Yi = Xi ) Cicm [Y ] = Cicm [X]81 / 88

7. Axiomatic characterization of the CMRS rule
I Theorem:
n

! L1+

n

I

Consider the RS rule C : L1+

I

C is the CMRS rule if and only if it satis…es the following
axioms:, with n

3.

(1) C is an aggregate RS rule.
(2) C satis…es no rip-o¤.
(3) C satis…es actuarial fairness.
(4) C satis…es coalition fairness.
I Proof:I
I) : See Denuit, Dhaene & Robert (2022).

( 22 : See Jiao, Kou, Liu & Wang (2022).

5

22 This is the hard part of the proof.
82 / 88

7. Axiomatic characterization of the CMRS rule

I Proposition23 :The axioms (1), (2), (3) and (4) are independent.
I Independence means that for any n

3 and any choice of 3of these axioms, there always exists a RS rulenn
C : L1+ ! L1+ which satis…es these 3 axioms, but not
the 4th axiom.

23 See J,K,L,W (2022).
83 / 88

7. Axiomatic characterization of the CMRS rule
I Theorem24 :nConsider the RS rule C : L1+

I

The RS rule C : L1+ ! L1+ is the CMRS rule if andonly if it satis…es the following axioms:

n

! L1+

n

I

, with n3.

n(1’) C is an aggregate RS rule.
(2’) C satis…es willingness-to-join.
(3’) C satis…es coalition fairness.
I Proof:I) : Follows immediately.

I( : Follows from property that ’willingness-to-join’implies’actuarial fairness’and ’no rip-o¤’.
5

24 See J,K,L,W (2022).
84 / 88

8. Further topics (not covered in this talk)I Quantile risk-sharing25 :I

The QRS rule is de…ned such that the contribution of eachparticipant i is given by FX 1 (p ).

I

Probability level p is determined such that the full allocation
condition is satis…ed.

i

I Pareto-optimal risk-sharing26 :I

What is the relation between Pareto-optimal risk-sharing andcomonotonicity?

I Risk-sharing pool dynamics27 :What if we add or withdraw one participant from the pool?28
What if the number of participants in the pool approaches
in…nity?
25 See D,D,R (2022) and D,R,C,D (2023).
26 see D,D,G,R (2023).
I

I

27 Work in progress.
28 See D,D,R (2022).
85 / 88References
I Borch K. (1962). Equilibrium in a reinsurance market. Econometrica

3, 424-444.
I Carlier G., Dana R.-A., Galichon A. (2012). Pareto e¢ ciency for the

concave order and multivariate comonotonicity. Journal of Economic
Theory 147, 207-229.
I Denneberg D. (1994). Non-additive measure and integral. Kluwer

Academic Publishers, Boston, pp. 184.
I Denuit M., Dhaene J. (2012). Convex order and comonotonic

conditional mean risk sharing. Insurance: Mathematics andEconomics, 51, 249-256.
I Denuit M., Dhaene J., Ghossoub M., Robert Y.C. (2023).

Comonotonicity and Pareto optimality, with application tocollaborative insurance. LIDAM Discussion Paper ISBA 2023/05.
I Denuit M. Dhaene J, Robert C.Y. (2022). Risk-sharing rules andtheir properties, with applications to peer-to-peer insurance. Journalof Risk and Insurance, 89(3), 615-667.
86 / 88References
I Dhaene J., Denuit M., Goovaerts M., Kaas R., Vyncke D. (2002a).

The concept of comonotonicity in actuarial science and …nance:
theory. Insurance: Mathematics & Economics, 31 (1), 3-33.
I Dhaene J., Denuit M., Goovaerts M.., Kaas R., Vyncke D. (2002b).

The concept of comonotonicity in actuarial science and …nance:
applications. Insurance: Mathematics & Economics, 31 (2),133-161.
I Dhaene J., Robert C.Y., Cheung K.C., Denuit M. (2023). An

axiomatic theory for quantile-based risk sharing. LIDAM Discussion
Paper ISBA, 2023/28.
I Feng R. (2023). Decentralized insurance. Springer Actuarial. pp.

263.
I Feng R., Liu M., Zhang (2022). A uni…ed theory of decentralized

insurance. SSRN.
I Jiao Z., Kou S., Liu Y, Wang R. (2022). An axiomatic theory foranonymized risk sharing. To be published.
87 / 88

Contact

I Jan Dhaene

Actuarial Research Group, KU Leuven
Naamsestraat 69, B-3000 Leuven, Belgium
I www.jandhaene.org
I jan.dhaene@kuleuven.be

88 / 88