A GAME THEORETIC MODEL FOR A REINSURANCE MARKET. SOME APPLICATIONS

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A GAME THEORETIC MODEL FOR A REINSURANCE MARKET.

SOME APPLICATIONS

MANUELA GHICA

Communicated by the former editorial board

We give a presentation of the essential and new elements useful in a reinsurance market applications of the game theory.We propose a structure for a game model applied in the case of a reinsurance market and then we define and give some applications with negative exponential and power utility for the market core, the market core cover, the reasonable and the Weber sets in a reinsurance market.

AMS 2010 Subject Classification: 62P05, 91B30, 90A12.

Key words: Reinsurance market, market core, market core cover, optimal allocations.

1. INTRODUCTION

We present in this paper an alternative model based on game theoretic approach. Cooperative game theory studies those situations where participants’

objectives are partially cooperative and partially conflicting. The biggest inte- rest of the participants is to cooperate because in this way they can achieve the greatest possible total benefits. When it comes to sharing the benefits of cooperation, however, individuals have conflicting goals. Such cases are usually modeled asn-person cooperative games in characteristic function form.

Many authors applied game theory ideas to the analysis of a reinsurance market, but recently, Aase [1–3], created some new connections between them finding the correspondences for “collective rationality” or Pareto optimality,

“social stability” and “individual rationality”. For other results focus on Pareto optimality relative to some optimization problems see for example [11–13]. The solutions of risk allocation problem end up in the core, concept introduced in the game theory by Gillies in 1959, [8]. Later, in 1963, Debreu and Scarf [5]

have proved that solution for the core in a market of pure exchange is nonempty and in 1981, Baton and Lemaire have found the core solution with the negative exponential utilities [4].

This paper begins with some preliminaries and the connections between the game theory and a reinsurance market. In Section 2 we include a brief

MATH. REPORTS15(65),2(2013), 161–176

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presentation for a model for a reinsurance market based on game theoretic approach and introduce some core catcher sets like market core cover, the Rea- sonable set and the Weber set for a reinsurance market. In the last section, we propose the solutions for the market core catcher sets in the case of the negative exponential utility function and in the case of the power utility function.

1.1. PRELIMINARIES

Let N be a group of n reinsurers, every one with preferences ≥_i, i∈ N, defined over a suitable set of random variables denoted byB, or gambles with realizations (outcomes) in some A ⊆ B. We represent these preferences by von Neumann-Morgenstern expected utility [10], meaning that there is a set of continuous utility functions u_i :B → R, such that X ≥_i Y if and only if Eui(X)≥Eui(Y),where by the symbolE we denoted the mean operator. We assume some properties: monotonic preferences and risk aversion.

We presume that each agent is invested with a random variable payoff X_i called initial portfolio. More precisely, there exists a probability space (Ω, K, P) such that we have the payoffXi(ω) whenω∈Ω occurs and, more, the both expected values and variances exist for all these initial portfolios, which means that allX_i∈L²(Ω, K, P).

Because every agent can treat any affordable contracts then we will have a new set of random variables Yi, i ∈ N, representing the final portfolios, Y_i∈L²(Ω, K, P).

The following notational convention will be used: if X and Y are two random variables, then by X ≤ Y we mean that Y −X ≥ 0 P-a.s., i.e., the random variable Y −X is nonnegative a.s..

In the following, we use the following notation: X_S:= P

i∈S

X_i,∀S ⊆N.

1.2. SOME CONNECTIONS BETWEEN THE GAME THEORY AND A REINSURANCE MARKET

A market of ideal exchange can be seen as a game. The players begin the game with an initial allocations of risks, or goods, exchange these risks (goods) in the market and finish with a final allocation which has a better utility.

Let N ={1,2, ..., n} be a set of agents and let S be an arbitrary subset of N. The characteristic function of the game v : 2^N → R gives the total payoff for the players who belong to the coalition S, S ∈2^N, payoff obtained by cooperating. Let z_i be the payoff to playeriwho cooperates in this game.

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The “collective rationality” corresponds to Pareto optimality in our reinsurance market

n

X

i=1

zi=v(N)⇔

n

X

i=1

λiEui(Yi) =Eu_λ_N(X_N)

where λN = (λ1, λ2, ..., λn), λi ∈ R^∗+, represent the agent weights and representative agent pricing denoted byX_N.

The next important condition from game theory: “individual rationality”

corresponds to Eui(Yi) ≥Eui(Xi), i∈ N, which implies that no player will participate in the game if he can obtain more alone.

In the situation of the rationality assumption for any coalition of all players we find the following condition: “social stability”

X

i∈S

z_i≥v(S), ∀S∈2^N ⇔X

i∈S

λ_iEu_i(Y_i)≥Eu_λ_S(X_S) where

Eu_λ_S(X_S) := sup

Zi,i∈S

X

i∈S

λ_iEu_i(Z_i) s.t. X

i∈S

Z_i≤X

i∈S

X_i:=X_S, a.s.

andλ_S = (λ_i₁, λ_i₂, ..., λ_i_s) = (λ_i)_i∈S, λ_i ∈R^∗+, S={i₁, ..., i_s}, S∈2^N,|S|=s.

The set of vectorsZ which satisfies the above relations is called the core of the game and it represents a very attractive solution when it exists but for a large class of games it is empty. This concept is very useful in economic applications.

2. A REPRESENTATION OF A MODEL AND SOME DEFINITIONS

In this section, we propose a structure for a game model applied in the case of a reinsurance market [7].

Definition 2.1.A competitive reinsurance market is a pair denoted by RM(u) =D

N, {Eu_λ_S(X_S)}_S⊆NE

consisting of the agent setN ={1,2, ..., n}

interpreted as (re)insurers where the function u_λ(·) :B →Ris von Neumann- Morgenstern expected utility function and Eu_λ_∅(X_∅) = 0.

LetRM(N, X) ={RM(u)|u∈ U }denote the set of the all reinsurance markets whereN ={1, 2, ..., n}is the set of the players,X= (X1, X2, ..., Xn) the initial random vectors,Xi∈L²(Ω,K, P)andU ={u|u:B→Ris concave and increasing.

Definition 2.2.A weight vector λ∈Rⁿ is animputation for the reinsurance market

D

N,{Eu_λ_S(X_S)}_S⊆NE

if it is efficient and it has the property of individual rationality, i.e.,

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1.

n

X

i=1

λiEui(Yi) =Eu_λ_N(XN) 2. Eui(Yi)≥Eui(Xi),∀i∈N.

We denote by I the set of imputations λ.Clearly,I is empty if and only if

n

X

i=1

λiEui(Yi)> EuλN(XN).We present in [7] the property for an essential reinsurance market where always there are infinitely many imputations.

Because the set of imputations is too large for an essential reinsurance market, so, we need some criteria to single out those imputations who have the chance to appear. So, we could obtain some subsets of I as solution concepts.

One of this solution concept is the core of a reinsurance market.

Definition 2.3. Themarket core denoted byM C of a reinsurance market D

is the set

(1) M C =

(

λ∈I|X

i∈S

λiEui(Yi)≥EuλS(XS),∀S⊆N )

If M C 6=∅then the elements ofM C can easily be obtained because the core is defined with the aid of a finite system of inequalities.

We introduce a set related on the market core namely: the market core cover. We can refer on it like a“market core catcher” because it contains the market core as a subset.

Definition 2.4.Let D

N,{Eu_λ_S(XS)}_S⊆NE

be a reinsurance market. For each i∈N and for each S ⊆N with i∈S the marginal contribution of agent ito the coalition S is

(2) Mi(S) =EuλS(XS)−EuλS\{i} XS\{i}

Definition 2.5.Let S⊆N, i∈S.The remainder R(S, i) of reinsurer i in the group of agents S is the amount which remains for reinsurer i if group of agents S forms and all other agents inS obtain their utopia payoffs, i.e.,

R(S, i) :=Eu_λ_S(X_S)−X M_j

j∈S\{i}

and for eachi∈N,thei-th coordinatem_i of the lower vectormis then defined by

(3) mi := max

S;S3iR(S, i)

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Definition 2.6. For a reinsurance marketD

the market core cover consists of all weight vectors for which we have

(4) M CC ={λ∈I|m_i ≤λ_iEu_i(Y_i)≤M_i, ∀i∈N} That M CC is a core catcher.

To describe a new kind of the set we will use the same notation: λS = (λi1, λi2, ...,λis) where we have the set S = {i₁, i2, ...,is} and |S| = s. In what it follows we present another market core catcher: the reasonable set.

This set was introduced in game theory by Milnor in 1952 [9].

Definition 2.7.The Reasonable set R of the reinsurance market denoted by D

is the setR= ∩

i∈NR_i,where, for∀i∈N, R_i has the following form:

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λ∈Rⁿ|λ_iEui(Xi)≤λiEui(Yi)≤ max

S;S3i

h

Eu_λ_S(XS)−Eu_λ_S\{i} X_S\{i}i Remark 2.1.Obviously M C ⊂M CC ⊂R.

The last market core catcher presented here correspond to the Weber set from game theory [15].

Definition 2.8. The Weber set W of the reinsurance market denoted by D

is the convex hull of the n! marginal vectors m^σ , corresponding to the n! permutations σ∈Π (N) :

m^σ_σ(1) = sup

Zσ(1) Zσ(1)≤Xσ(1)

λ_σ(1)Eu_σ(1) Z_σ(1)

m^σ_σ(2) = sup

Zσ(1),Zσ(2) Zσ(1)+Zσ(2)≤Xσ(1)+Xσ(2)

Euλ{σ(1),σ(2)} Z{σ(1),σ(2)

− sup

Zσ(1) Zσ(1)≤Xσ(1)

λ_σ(1)Eu_σ(1)Z_σ(1)

...

m^σ_σ(k) = sup

Zσ(1),..,Zσ(k)

Zσ(1)+...+Zσ(k)≤Xσ(1)+...+Xσ(k)

Eu_λ{σ(1),...,σ(k) Z{σ(1),...,σ(k)

− sup

Zσ(1),..,Zσ(k−1)

Zσ(1)+...+Zσ(k−1)≤Xσ(1)+...+Xσ(k−1)

Eu_λ{σ(1),...,σ(k−1) Z{σ(1),...,σ(k−1)

for eachk∈N.

The payoff vector m^σ can be created as follows. Let the agents enter in a room one by one in the order: σ(1), ..., σ(n) and give each agent the marginal contribution which he creates by entering.

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3.SOME APPLICATIONS

3.1. THE EXPONENTIAL UTILITY FUNCTION

In general the core will be characterized by the Pareto optimal allocations according to investor weightsλ_i in some region restricted by inequalities given in the following theorems.

The initial portfolios are denoted byX1, X2, .., Xn, the “market portfolio”

is XN =

n

X

i=1

Xi and the reinsurers have the exponential utility functions given by: u_i(x) = 1−a_ie⁻

x

ai, x∈R, i∈N.

From [1, 2] The Pareto optimal allocations who result from coalition are Y_i(X_N) = ^a_AⁱX_N +b_i(N),whereb_i(N) =a_ilnλ_i−a_i^K_A, A=

n

X

i=1

a_i and K=

n

X

i=1

a_ilnλ_i

and for any subset S ⊆N the formulas from above becomes:

Y_i(X_S) = _A^aⁱ

SX_S+b_i(S),whereb_i(S) =a_ilnλ_i−a_i^K_A^S

S, A_S = P

i∈S

a_i, K_S= P

i∈S

a_ilnλ_i, X_S= P

i∈S

X_i

The values λ_i appear as “investor weights” are arbitrary positive constants and we want to find the constraints for the values sets of these constants, or equivalently, to deceive constraints on the zero-sum side paymentsbi.

We use for these requirements the expected utility of the representative agents for any S⊆N :

EuλS(XS) =E

P

i∈S

λi−ASe

KS−XS AS

Theorem 3.1.1 ([2]). In case of the reinsurance market with n = 3 agents, the market core will be characterized by the Pareto optimal allocations corresponding to investor weights λ_i, who are solutions of the system of inequalities:

λ_i ≥

E

e⁻^XN^A

E

"

e⁻

Xiai

#e^K^A

A_Sd_i,jE

e⁻^XS^AS

≥a_iEh e⁻^XN^A i

+a_jEh e⁻^XN^A i

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2AEh e⁻^XN^A i

≤ P

i,j∈{1,2,3},i6=j

A_Sd_i,jE

e⁻

Xi+Xj ai+aj

b₁(N) +b₂(N) +b₃(N) = 0

i, j∈ {1, 2, 3}, i6=j, S⊂N,where we denoted by di,j =e

bi+bj ai+aj

.

We can see that the market core is characterized by the Pareto optimal allocations corresponding to investor weights λ_i in some region restricted by inequalities of the above kind, in general a polyhedron from int Rⁿ₊

.

Theorem 3.1.2. The Pareto optimal allocations which correspond to the market core cover are given by weights λ_i, i∈ N, the solutions of the system of next inequalities:

α_i ≤ λ_i ≤β_i, i∈N, where

α_i = e^bi

(N) ai Eh

e^A¹^X^Ni

C_N −C_N\{i}] βi = e^bi

(N) ai E

h e^A¹^X^N

i







S;minS3i



(|S| −2)CS− X

j∈S−{i}

CS\{j}











i∈N,where e^KS

−XS

AS =B_S(X) andA_S·E[B_S(X)] =C_S

Proof. From definition of the market core cover we have the double inequalities (4):

m_i ≤ λ_iEu_i(Y_i) ≤ M_i,∀i ∈ N and for the utility function u_i(x) = 1−aie⁻

x

ai these inequalities become mi ≤λiE

1−e⁻^Yi^ai

≤Mi,∀i∈N.

The first term of inequalities mi,defined in (3), becomes:

mi = max

S;S3iR(S, i) = max

S;S3i

"

EuλS(XS)− P Mj j∈S−{i}

#

= max

S;S3i

"

EuλS(XS)− P

j∈S\{i}

h

EuλS(XS)−EuλS\{j} XS\{j}

i

#

= max

S;S3i

( E

P

i∈S

λi−ASBS(X)

− P

j∈S\{i}

λj−CS + C_S\{j}

)

= max

S;S3i

( P

i∈S

λ_i−C_S− P

j∈S\{i}

λ_j−C_S+C_S\{j}

)

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= max

S;S3i

(

λ_i−C_S+ (|S| −1)C_S− P

j∈S\{i}

C_S\{j}

)

= λi+ max

S;S3i

(

(|S| −2)CS− P

j∈S\{i}

C_S\{j}

) .

And the last term of the inequalities, defined in (2), becomes:

M_i =Eu_λ_N(X_N)−Eu_λ_N\{i} X_N\{i}

=

=E

P

i∈N

λi−AB_N(X)

−E P

j∈N\{i}

λj−A_N\{i}B_S\{i}(X)

!

=

= P

i∈N

λ_i−AE[B_N(X)]− P

j∈N\{i}

λ_j+A_N\{i}E[B_S\{i}(X)] =

=λ_i−AE[B_N(X)] +A_N_\{i}E[B_S\{i}(X)]

So, the theorem inequalities become:

S;maxS3i

"

(|S| −2)A_SE[B_S(X)]− P

j∈S\{i}

A_S\{j}E

B_S\{j}(X)

#

≤

≤ −λ_iE

e⁻^Yi^ai

⇒

−λ_iE

e⁻^Yi^ai

≤ −AE[B_N(X)] +A_N\{i}E[B_S\{i}(X)]

and the last form is:

e^bi^aiEh e^A¹^X^Ni

AE[B_N(X)]−A_N_\{i}E[B_S\{i}(X)]

≤ λ_i λ_i ≤ e^ai^biEh

e^A¹^X^Ni

S;minS3i

(

(|S| −2)C_S− P

j∈S\{i}

C_S\{j}

) .

Theorem 3.1.3. The Pareto optimal allocations which correspond to a reasonable set R for a reinsurance market

D

are given by weights λ_i, i∈N, the solutions of the system of inequalities:

λ_i≥max









 Eh

e⁻^XN^A i

E

e⁻^Xi^ai

e^K^A , a⁻¹_i E(e⁻

1

AXN−^bi^(N)

ai ) S;maxS3i

C_S+C_S\{i}











i∈N, A_SE[B_S(X)] =C_S and e^KS

−XS

AS =B_S(X),∀S ⊆N.

Proof. From relation (5) the reasonable set is the set given∀i∈N by

λ∈Rⁿ|λ_iEu_i(X_i)≤λ_iEu_i(Y_i)≤ max

S;S3i

h

Eu_λ_S(X_S)−Eu_λ_S\{i} X_S\{i}i .

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So, for the utility function given above we will have:

λ_iE(1−a_ie⁻^Xi^ai)≤λ_iE(1−a_ie⁻^Yi^ai)≤

≤ max

S;S3i

"

E

P

i∈S

λ_i−A_Se^KS

−XS AS

−E P

j∈S\{i}

λ_j−A_S\{i}e

KS\{i}−XS\{i}

AS\{i}

!#

and equivalent we obtain:

λ_i−a_iλ_iE(e⁻^Xi^ai)≤λ_i−a_iλ_iE(e⁻

1

ai(^ai_A^XN+bi(N))) and λi−aiλiE(e⁻

1

ai(^aiAXN+bi(N)))≤

≤ max

S;S3i

"

λi−ASE(e

KS−XS

AS )−AS\{i}Ee

KS\{i}−XS\{i}

AS\{i}

# . So, the above inequalities becomes:

e⁻^bi

(N) ai ≤

E

"

e⁻

Xiai

#

E

e⁻^XN^A and

S;maxS3i

"

A_SE(e^KS

−XS

AS ) +A_S\{i}Ee

KS\{i}−XS\{i}

AS\{i}

#

≤a_iλ_iE(e

1 AXN+^bi

ai) or,

λi ≥

E

e⁻^XN^A

E

"

e⁻^Xi^ai

#e^K^A and

a⁻¹_i E(e⁻

1

AXN−^bi^(N)

ai ) max

S;S3i

"

A_SE(e^KS

−XS

AS )+A_S\{i}Ee

KS\{i}−XS\{i}

AS\{i}

#

≤λ_i. Remark 3.1.1. The Pareto optimal allocations who correspond to a Weber setW for a reinsurance marketD

are given by the convex hull of the n! marginal vectorsm^σ, corresponding to then! permutationsσ ∈ Π (N) :

m^σ_σ(1) =λ_σ(1)E(1−a_σ(1)e⁻

Xσ(1) aσ(1)) m^σ_σ(2) =E

"

λ_σ(1)+λ_σ(2)− a_σ(1)+a_σ(2) e

aσ(1) lnλσ(1)+aσ(2) lnλσ(2)−(Xσ(1)+Xσ(2)) aσ(1)+aσ(2)

#

− –λ_σ(1)E(1−a_σ(1)e⁻

Xσ(1) aσ(1)) =

=λ_σ(2)+E

"

a_σ(1)e⁻

Xσ(1)

aσ(1) − a_σ(1)+a_σ(2) e

# . ...

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m^σ_σ(k)=Eu_λ{σ(1),...,σ(k) X{σ(1),...,σ(k)

−Eu_λ{σ(1),...,σ(k) X{σ(1),...,σ(k−1)}

= –λ_σ(k)+

E

"

Ak−1e

K{σ(1),...,σ(k−1)}−X{σ(1),...,σ(k−1)}

A{σ(1),...,σ(k−1)} −A_k e

K{σ(1),...,σ(k)}−X{σ(1),...,σ(k)}

A{σ(1),...,σ(k)}

# for each k∈N

where we denote by Ak−1 =A{σ(1),...,σ(k−1)}and Ak=A{σ(1),...,σ(k)}.

3.2. THE POWER UTILITY FUNCTION In this case we consider the utility function ui(x) = ^x¹⁻_1−a^ai⁻¹

i for x > 0, a_i 6= 1; for situation when x > 0, a_i → 1 we have the case of the natural logarithmic utility u_i(x) = ln (x).

This kind of power utility function only make sense in the no-bankruptcy case whereXi>0 a.s. for alli.The parameters ai >0 are interpreted like the relative risk aversions of the agents.

We take the case where a₁ =a₂ =...=a_n=a. So, the marginal utilities u⁰_i(x) =x^−a and using the Riesz representation [14] we get

u⁰_i(Y_i(X_N)) =α_iξ(X_N) a.s. for alli which implies a formula for the Pareto optimal allocations:

Yi =α⁻

1 a

i ξ⁻¹^a(X_N), a.s.

and if we use the clearing market (

n

X

i=1

Yi =X_N) we get

ξ(X_N) =

n

X

i=1

λ

1 a

i

!a

X_N^−a, where λ_i = 1 α_i In the end we get that the optimal sharing rules are linear:

Yi(XN) = λ

1 a

i n

X

i=1

λ

1 a

i

XN a.s. for all i

and for any subset S ⊆N the formulas from above become:

Y_i(X_S) = λ

1 a

i

P

i∈S

λ

1 a

i

X_S

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The weights λ_i can be determined by the condition of the budget constraints: E[Y_iξ] =E[X_iξ].In our case of the power utility functions we get

λ_i=

n

X

i=1

λ

1 a

i

!a"

E

X_iX_N^−a E

X_N^1−a

#

a.s. for all i If we normalize such that E[ξ(XN)] = 1 then we get that

n

X

i=1

λ

1 a

i

!a

= 1

E X_N^−a

From the last two equations we get the formula for the weights λ_i : λi = 1

E X_N^−a

"

E

X_iX_N^−a E

X_N^1−a

#

With this results we can find, also, the market price:

π(Z) =E[Zξ] = E

ZX_N^−a E

X_N^−a , for all Z ∈L²(Ω, K, P)

Notice that we have also found the characteristic function of the game, here given by the expected utility of the “representative agent” restricted to any subsetS ⊆N.

Eu_λ_S(X_S) =E







P

i∈S

λ

1 a

i

a

X_S^1−a−P

i∈S

λi

1−a







Theorem3.2.1. In case of the reinsurance market withn= 3 agents, the market core will be characterized by the Pareto optimal allocations corresponding to investor weights λ_i, i∈N, the solutions of the system of inequalities:

λi ≥ ¹

E[^XN^−a]

E[^Xi^1−a]

E[^XN^1−a] _1−a^a

, i∈ {1,2,3}

λ_i ≥2^−a

3

X

i=1

λ

1 a

i

!a

[a_i,j+a_i,k−a_k,j]^a,i, j, k ∈ {1,2,3}, i6=j6=k, i6=k

2

3

X

i=1

λ

1 a

i

!a

≥ X

{i,j}∈{{1,2},{1,3},{2,3}}

λ

1 a

i +λ

1 a

j

a

a^1−a_i,j

where ai,j = h_E[X

i+Xj]^1−a E[XN]^1−a

i_1−a¹

for {i, j} ∈ {{1,2},{1,3},{2,3}}.

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The core allocations Y = (Y₁, Y₂, ..., Y_n) are given by Y_i(X_N) = λ

1 a

i n

X

i=1

λ

1 a

i

X_N a.s. for all i

and λi are defined by the inequalities proved above.

Theorem 3.2.2. The Pareto optimal allocations which correspond to the market core cover are given by weights λi i∈N, the solutions of the system of next inequalities:

λ_i ≥ ¹

E[^XN^1−a]max

S;S3i

"

(2− |S|)λ_a,SE X_S^1−a

+ P

j∈S−{i}

λ_a,S\{j}Eh

X_S\{j}^1−a i

#

λ_i ≤

E[^X^1−aN ]

E[^X^−aN ]^−λ^a,N\{i}^E

h X_N\{i}^1−a i E[^X_N^1−a]

where

P

i∈S

λ

1 a

i

a

=λ_a,S,∀S ⊆N.

Proof. From definition of the market core cover we have the double inequalities (4):

m_i≤λ_iEu_i(Y_i)≤M_i,∀i∈N and for the utility functionu_i(x) = ^x^1−a_1−a⁻¹ forx >0, a6= 1

these inequalities become m_i ≤λ_iEh_Y1−a i −1

1−a

i≤M_i,∀i∈N.

The first term of inequalities mi,defined in (3), becomes:

m_i = max

S;S3iR(S, i) = max

S;S3i

"

Eu_λ_S(X_S)− P M_j

j∈S−{i}

#

= max

S;S3i

"

Eu_λ_S(X_S)− P

j∈S−{i}

h

Eu_λ_S(X_S)−Eu_λ_S\{j} X_S\{j}i

#

= max

S;S3i







λa,SE[^X_S^1−a]⁻^P

i∈S

λi

1−a −

P

j∈S−{i}

h

−λj+λa,SE[^X_S^1−a]^−λa,S\{j}Eh

X_S\{j}^1−a ii 1−a







=

−λ_i+ max

S;S3i

"

λa,SE[^XS^1−a]^{−(|S|−1)λ}a,SE[^XS^1−a]⁺ ^P

j∈S−{i}

λa,S\{j}E h

X_S\{j}^1−a i

#

1−a

=

−λ_i+ max

S;S3i

"

(2−|S|)λ_a,SE[^X^1−aS ]⁺ ^P

j∈S−{i}

λa,S\{j}E h

X_S\{j}^1−a i

#

1−a

=-_1−a^λⁱ +_1−a¹ max

S;S3i

"

(2− |S|)λ_a,SE X_S^1−a

+ P

j∈S−{i}

λ_a,S\{j}Eh

X_S\{j}^1−a i

#

(13)

where

P

i∈S

λ

1 a

i

a

=λa,S.

And the last term M_i, defined in (2), becomes:

M_i =Eu_λ_N(X_N)−Eu_λ_N\{i} X_N\{i}

=

=_1−a¹

E[^XN^1−a]

E[^XN^−a] −Pn i=1λ_i

−_1−a¹

"

λ_a,N_\{i}Eh

X_N\{i}^1−a i

− P

j∈N−{i}

λ_j

#

=-_1−a^λⁱ +_1−a¹

E[^XN^1−a]

E[^X_N^−a] −λ_a,N_\{i}Eh

X_N\{i}^1−a i

So, first inequality is -_1−a^λⁱ +_1−a¹ max

S;S3i

"

(2− |S|)λa,SE X_S^1−a

+ P

j∈S−{i}

λa,S\{j}E h

X_S\{j}^1−a i

#

≤

≤λ_iEh_Y^1−a

i −1 1−a

i

and the second inequality becomes λ_iE

Y_i^1−a−1 1−a

≤ − λ_i

1−a+ 1 1−a

E X_N^1−a E

X_N^−a −λ_a,N\{i}Eh

X_N\{i}^1−a i

! .

In the end we obtain:

1

E[^XN^1−a]max

S;S3i

"

(2− |S|)λ_a,SE X_S^1−a

+ P

j∈S−{i}

λ_a,S\{j}Eh

X_S\{j}^1−a i

#

≤λ_i≤

E[^X^1−aN ]

E[^X^−aN ]^−λ^a,N\{i}^E

h X_N\{i}^1−a

i

E[^XN^1−a] .

Theorem 3.2.3. The Pareto optimal allocations which correspond to a reasonable set R for a reinsurance market

D

aree given by weights λ_i i∈N,the solutions of the system of inequalities:

λ_i ≥max{R₁(X), R₂(X)}

with

R_1,i(X) = E X_N^1−a E

X_i^1−aE[X_N^a] and

R2,i(X) =

S;maxS3i

h

λ_a,SE X_S^1−a

−λ_a,S\{i}Eh

X_S\{i}^1−a iia

E X_N^1−a

E[XN]^a .

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Proof. From relation (5) the reasonable set is the intersection of the sets given by

λ∈Rⁿ|λ_iEu_i(X_i)≤λ_iEu_i(Y_i)≤max

S;S3i

h

Eu_λ_S(X_S)−Eu_λ_S\{i} X_S\{i}i ,∀i∈N

First inequality from the reasonable set is the individual rationality Eui(Yi)≥Eui(Xi), i∈N.

This is equivalent with λiE

X_i^1−a−1 1−a

≤λiE

Y_i^1−a−1 1−a

⇔

E X_i^1−a

≤E











 λ

1 a

i n

X

i=1

λ

1 a

i

X_N







1−a





⇔

E X_i^1−a

≤E





 λ

1−a a

i n

X

i=1

λ

1 a

i

!1−aX_N^1−a







⇔

λi ≤ E X_N^1−a E

X_i^1−aE[X_N^a].

So, for the utility function given above, the second inequality from the reasonable set becomes:

λiEui(Yi)≤ max

S;S3iE







P

i∈S

λ

1a i

!a

X_S^1−a−P

i∈S

λi− P

S\{i}

λ

1a j

!a

X_S\{i}^1−a − P

S\{i}

λj

1−a





⇔ λiE

Y_i^1−a

≤ max

S;S3i

n

λa,SE X_S^1−a

−λ_a,S\{i}E h

X_S\{i}^1−a io

⇔

λi ≤

max

S;S3i

h

λa,SE[^X_S^1−a]^−λa,S\{i}Eh

X_S\{i}^1−a iia

E[^XN^1−a]^E[XN]^a ,where

P

i∈S

λ

1 a

i

a

=λa,S. Remark 3.2.1. The Pareto optimal allocations who correspond to a Weber setW for a reinsurance marketD

are given by the convex

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hull of the n! marginal vectorsm^σ, corresponding to then! permutationsσ ∈ Π (N) :

m^σ_σ(1) = ^λ_1−a^σ(1)E(X_σ(1)^1−a−1) m^σ_σ(2) =E

"

λ_σ(1)+λ_σ(2)− a_σ(1)+a_σ(2) e

#

− –λ_σ(1)E(1−a_σ(1)e⁻

Xσ(1) aσ(1)) =

=−^λ_1−a^σ(2) + _1−a¹ Eh

λ^1/a_σ(1)+λ^1/a_σ(2)a

X_σ(1)+X_σ(2)1−a

−λ_σ(1)X_σ(1)^1−ai ...

m^σ_σ(k)=Eu_λ X{σ(1),...,σ(k)

−Eu_λ X{σ(1),...,σ(k−1)}

=

−λ_σ(k)+E

λa,{σ(1),..σ(k)}(^Xσ(1)+...+X_σ(k))^1−a^−λ_a,{σ(1),..σ(k−1)}(^Xσ(1)+...+X_σ(k−1))^1−a

1−a

for each k∈N, whereλ_a,{σ(1),.._σ(k)}=



 P

i∈{σ(1),.._σ(k)} λ

1 a

i





a

.

REFERENCES

[1] K.K. Aase, Equilibrium in a reinsurance syndicate; Existence, uniqueness and charac- terization. ASTIN Bulletin22(1993), 185-21.

[2] K.K. Aase,Perspectives of Risk Sharing, Scand. Actuarial Journal2(2002), 73–128.

[3] K.K. Aase, Optimal Risk Sharing. In J.L. Teugels& B. Sundt (Eds.) Encyclopedia of actuarial science3(2004) 1212–1215, Chichester, UK: Wiley.

[4] B. Baton and J. Lemaire,The Core of a Reinsurance Market.ASTIN Bulletin12(1981), 1,57–71.

[5] G. Debreu and H. Scarf,A Limit Theorem on the Core of an Economy. International Economic Review4(1963), 235–246.

[6] H. Gerber, On Additive Premium Calculation Principles. ASTIN Bulletin 7 (1974), 215–222.

[7] M. Ghica,The Core of a Reinsurance Market. Mathematical Reports10(60)(2008),2, 155–164.

[8] D.B. Gillies,Solutions to General Nonzero-Sum Games. Annals of Mathematical Stud- ies,40(1959), 47–85.

[9] J.W. Milnor,Reasonable Outcomes for n-person Games.Research Memorandum RM916, The RAND Coorporation, Santa Monica, 1952.

[10] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behaviour.

Princeton University Press, 1944.

[11] V. Preda,On Duality with Generalized Convexity. Bolletino della Unione Matematica Italiana,5A(1991),3,291–305.

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[12] V. Preda,On Efficiency and Duality for Multiobjective Programs. Journal of the Math- ematical Analysis and Applications,166(1992),2,365–377.

[13] V. Preda,Optimality and Duality in Fractional Multiple Objective Programming Involv- ing Semilocally Preinvex and Related Functions. Journal of the Mathematical Analysis and Applications,288(2003),2,365–382.

[14] T. Rockafellar,Convex Analysis. Princeton University Press, Princeton, N.Y., 1970.

[15] R. Weber, Probabilistic Values for Games. In: Roth, A.E. (Ed.), The Shapley Value:

Essays in Honour of Lloyd S. Shapley, Cambridge University Press, 101–119, 1988.

Received 6 March 2012 Spiru Haret University

Faculty of Mathematics and Informatics Str. Ion Ghica 13

030045 Bucharest, Romania