A GAME THEORETIC MODEL FOR A REINSURANCE MARKET. SOME THEORETICAL RESULTS

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FOR A REINSURANCE MARKET.

SOME THEORETICAL RESULTS

MANUELA GHICA

Communicated by the former editorial board

We propose a game-theoretic model for a reinsurance market. We define three market core catcher: the market core cover, the reasonable set, and the Weber set for a reinsurance market. Some properties are given.

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. Many authors applied game theory ideas to the analysis of a reinsurance market, but recently Aase, [1], [2], [3], created some new connections, between them finding the correspondences for “collective rationality”, “social stability” and “individual rationality”. The solutions of risk allocation pro- blem 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 the 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 utili- ties [4]. For Pareto optimality relative to some optimization problems see for examples [11], [12], [13].

This paper begins with some preliminaries and the connections between the game theory and a reinsurance market. In Section 2, we include a brief presentation for a model of a reinsurance market based on game theoretic approach. We introduce in the last section some core catcher sets like market core cover, the reasonable set and the Weber set for a reinsurance market and we present some properties.

MATH. REPORTS15(65),1 (2013), 79–89

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1.1. PRELIMINARIES

LetN be a group ofn 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 symbol E we denoted the mean operator.

We assume some properties: monotonic preferences and risk aversion, so that, we have u⁰_i(w) >0, u⁰⁰_i(w) ≤0, for all w in the relevant domains. In some of the cases we shall also require strict risk aversion, meaning strict concavity for some u_i.We presume that each agent is invested with a random variable payoffXicalledinitial portfolio. More precisely, there exists a probability space (Ω,K, P) such that we have the payoff X_i(ω) whenω ∈Ω occurs and, more, the both expected values and variances exist for all these initial portfolios, which means that all X_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 thefinal 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 initial allocations of risks, or goods, exchange these risks in the market and finish with a final portfolio which has a better utility.

LetN ={1,2, . . . , n} be a set of agents and letS 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 playeri who cooperates in this game.

The “collective rationality” or the fact that the players who will obtain by cooperating the maximum total payoff can be represented in this way:

n

X

i=1

z_i =v(N).

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This assumption corresponds to Pareto optimality in our reinsurance market, i.e., the optimal solution Y solver

n

X

i=1

λiEui(Yi) =EuλN(XN),

where λ_N = (λ₁, λ₂, . . . , λ_n), λ_i ∈R^∗+,represent the agent weights and repre- sentative agent pricing denoted by XN.

The last important condition z_i ≥ v({i}) represents “individual rationality” and corresponds to Eui(Yi)≥Eui(Xi), i∈N,which implies that no player will participate in the game if he can obtain more alone.

If we impose the rationality assumption for any coalition of all players, i.e., for any S ∈2^N,we have

X

i∈S

z_i ≥v(S), ∀S∈2^N.

We can name this condition: “social stability” and it corresponds in reinsurance market to a further restriction on the investor weights λ 6= 0 such that

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.

and λ_S = (λi1, λi2, . . . , λis) = (λi)_i∈S, λi ∈ R^∗+, S = {i₁, . . . , is}, S ∈ 2^N,

|S|=s.

We see that forS={i},|S|= 1, we have Eu_λ_{i} X_{i}

=λiEui(Xi).

The set of vectorsZ which satisfies the above relation 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

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) =hN,{Eu_λ_S(XS)}_S⊆Niconsisting of the agent setN ={1,2, . . . , n}

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

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

In the next part we denote the set of “investor weights”λ_i,i= 1, n,by

I^∗∗=

λ∈Rⁿ|

n

X

i=1

λiEui(Yi)≤EuλN(XN)

.

Here λiEui(Yi) =Euλ_{i} Y{i}

. And by I^∗ the set of efficient “investor weights” vectors in the reinsurance market hN,{Eu_λ(X_S)}_S⊆Ni, i.e.,

I^∗ =

λ∈Rⁿ|

n

X

i=1

λiEui(Yi) =Euλ_N(XN)

.

Obviously, we have I^∗ ⊂I^∗∗.

The “individual rationality” conditionEui(Yi)≥Eui(Xi), i∈N should hold in order that a weight vector λ has a real chance to be realized in the reinsurance market.

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

N,

EuλS(XS) _S⊆N

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

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n

P

i=1

λiEui(Yi) =Eu_λ_N(XN);

(2)Eu_i(Y_i)≥Eu_i(X_i), ∀i∈N.

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

n

P

i=1

λ_iEu_i(Y_i) > Eu_λ_N(X_N).We present in [7] the property for an essential reinsurance market where there are always infinitely many imputations.

3. SOME SETS RELATED WITH MARKET CORE FOR A REINSURANCE MARKET

Because the set of imputations is too large for an essential reinsurance market, 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 these solution concepts is the core of a reinsurance market.

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Definition 3.1. Themarket coredenoted byM C of a reinsurance market hN,{Eu_λ_S(X_S)}_S⊆Niis the set

(1) M C =

λ∈I |X

i∈S

λiEui(Yi)≥Euλ_S(XS), ∀S ⊆N

.

If M C 6= ∅ then, the elements of M 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.

We denote withM the upper allocation and withm the lower allocation which is important for defining the new element. For each i ∈ N, the ith coordinate Mi of the upper vector M is the marginal contribution of the reinsurer iin the grand group of agents; it is also called the utopia payoff for reinsurer i in the grand group of agents in the sense that if the reinsurer i wants more then it is advantageous for the other agents in N to throw agent i out.

Definition 3.2. Let hN,{Eu_λ_S(XS)}_S⊆Ni be a reinsurance market. For each i∈N and for eachS⊆N withi∈S the marginal contribution of agent i to the coalitionS is

Mi(S) =EuλS(XS)−Euλ_S\{i} XS\{i}

.

Definition 3.3. Let S⊆N, i∈S.The remainder R(S, i) of reinsureriin the group of agentsS is the amount which remains for reinsureriif the 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 each i ∈ N, the ith coordinate mi of the lower vector m is then defined by

mi := max

S;S3iR(S, i).

Definition 3.4. For a reinsurance markethN,{Eu_λ_S(XS)}_S⊆Nithe market core cover consists of all weight vectors for which we have

M CC ={λ∈I |m_i ≤λ_iEu_i(Y_i)≤M_i,∀i∈N}.

ThatM CCis a core catcher follows from the next theorem which presents that the lower (upper) vector is a lower (upper) bound for the market core.

Theorem 3.1. Let hN,{Eu_λ_S(X_S)}_S⊆Ni be a reinsurance market and λ∈M C. Then, mi ≤λiEui(Yi)≤Mi,∀i∈N.

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Proof. (i)λ_iEu_i(Y_i) = Pn

i=1

λ_iEu_i(Y_i)− P

j∈N\{i}

λ_jEu_j(Y_j) =Eu_λ_N(X_N)− P

j∈N\{i}

λjEuj(Yj)≤Eu_λ_N(XN)−Eu_λ_N\{i} X_N\{i}

=Mi(N), for eachi∈N.

(ii) In view of (i), for each S ⊆N and eachi∈S we have λ_iEu_i(Y_i) = P

i∈S

λ_iEu_i(Y_i)− P

j∈N\{i}

λ_jEu_j(Y_j) ≥ Eu_λ_S(X_S)− P

j∈S\{i}

M_j(N) = R(S, i).

So, λiEui(Yi)≥ max

S;S3iR(S, i) =mi, for each i∈S.

To describe a new kind of the set we will use the same notation: λ_S = (λi1, λi2, . . . , λis) where we have the set S = {i₁, i2, . . . , is} and |S| = s. In what follows we present another market core catcher: the reasonable set. This set was introduced in game theory by Milnor in 1952, [5].

Definition 3.5. The reasonable set R of the reinsurance market denoted by hN,{Eu_λ_S(X_S)}S⊆Ni is the set R = T

i∈N

R_i where, for ∀i∈N, R_i has the next form

λ∈Rⁿ|λiEui(Xi)≤λiEui(Yi)≤ max

S;S3i

h

Euλ_S(XS)−Euλ_S\{i} XS\{i}

i .

Remark 3.1. Obviously, M C ⊂M CC ⊂R.

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

Definition 3.6. The Π(N)-Weber set of the reinsurance market hN,{E uλS(XS)}_S⊆Ni, denoted by W(Π(N)), is the convex hull of the n! marginal vectors m^σ, where

m^σ(λ^σ) =

m^σ_σ(1) λ_σ₍₁₎

, m^σ_σ(2) λ{σ(1),σ(2)}

, . . . , m^σ_σ(n) λ{σ(1),σ(2),...,σ(n)}

corresponding to the n! permutationsσ∈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) ...

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m^σ_σ(k)= sup

Zσ(1),...,Zσ(k)

Zσ(1)+···+Zσ(k)≤Xσ(1)+···+Xσ(k)

Euλ{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 each k∈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.

Definition 3.7. TheWeber set of the reinsurance market hN,{Eu_λ_S(X_S)}_S⊆Ni

is the set

W ={λ∈I if∃σ ∈Π(N) :m^σ(λ^σ)∈W (Π(N))}.

We propose for proving the following result which is necessary to show that M C ⊂W.

Proposition 3.1. The set M C is a convex set.

Proof. Let be ˜λ,¯λ∈M C. From (1) we have that:

(1) P

i∈S

λ˜_iEu_i(Y_i)≥Eu_˜_λ

S(X_S);

(2) P

i∈S

λ¯iEui(Yi)≥Eu¯λS(XS).

If we multiply the first relation with α and the second relation with (1−α),by summing we obtain

X

i∈S

h

αλ˜i+ (1−α) ¯λi

i

Eui(Yi)≥αEu˜λS(XS) + (1−α)Eu¯λS(XS).

Since, Eu_˜_λ

S(X_S) = sup

P

i∈S

Zi≤P

i∈S

Xi

P

i∈S

λ˜_iEu_i(Z_i) and Eu¯λS(X_S) = sup

P

i∈S

Zi≤P

i∈S

Xi

P

i∈S

λ¯_iEu_i(Z_i) the last term of the above inequality becomes

α sup

P

i∈S

Zi≤P

i∈S

Xi

X

i∈S

˜λiEui(Zi) + (1−α) sup

P

i∈S

Zi≤P

i∈S

Xi

X

i∈S

¯λiEui(Zi) =

= sup

P

i∈S

Zi≤P

i∈S

Xi

αX

i∈S

λ˜iEui(Zi) + sup

P

i∈S

Zi≤P

i∈S

Xi

(1−α)X

i∈S

λ¯iEui(Zi)≥

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≥ sup

P

i∈S

Zi≤P

i∈S

Xi

"

αX

i∈S

λ˜_iEu_i(Z_i) + (1−α)X

i∈S

λ¯_iEu_i(Z_i)

#

=

=E h

αuλ˜S(XS) + (1−α)uλ¯S(XS) i

. We proved that

X

i∈S

h

αλ˜i+ (1−α) ¯λi

i

Eui(Yi)≥E h

αu˜λS(XS) + (1−α)uλ¯S(XS) i

, so, we have that αλ˜+ (1−α) ¯λ∈M C.

The Weber set is a market core catcher as shown in the next theorem.

We will prove that M C ⊂W.

Theorem 3.2. Let hN,{Eu_λ_S(XS)}_S⊆Ni be the reinsurance market.

Then, M Cg ⊂W(Π(N)) where M Cg is

(λ₁Eu₁(Y₁), . . . , λ_nEu_n(Y_n))|X

i∈S

λ_iEu_i(Y_i)≥Eu_λ_S(X_S),∀i∈S, λ∈I

.

Proof. If|N|= 1,then M Cg =W (Π(N)) ={λ₁Eu₁(X₁)}. If|N|= 2,we have two cases: I =∅and I 6=∅.

For I = ∅ then M Cg = ∅ ⊂ W (Π(N)). Another case for I 6= ∅ we consider

x⁰ =

Eu_λ_{1} X_{1}

, Eu_λ_{1,2} X_{1,2}

−Eu_λ_{1} X_{1}

and

x⁰⁰=

Eu_λ{₂_} X_{2}

, Eu_λ_{1,2} X_{1,2}

−Eu_λ{₂_} X_{2}

and we see thatM Cg =co{x⁰, x⁰⁰}=co{m^σ |σ :{1,2} → {1,2}}=W (Π(N)). We go forth by induction of the number of agents. So, we suppose|N|= n > 2 and the set M Cg is a subset of the Π(N)-Weber set, for all reinsurance market with number of agents smaller than n. TheM Cg andW (Π(N)) are convex sets. Then, from the characterization theorem of extreme points of a polyhedral set [14] it follows that there exists T ∈ 2^N\ {∅, N} with

P

i∈T

λiEui(Yi) =Eu_λ_T(XT). Let us consider RM(¯u) = hT,{Eu¯_λ_S(XS)}_S⊆Ti and RM(˜u) =hN\T,{Eu˜_λ_S(X_S)}_S⊆N\Tidefined by

Eu¯_λ_S(X_S) =Eu_λ_S(X_S), S ∈2^T,

Eu˜_λ_S(X_S) =Eu_λ_T∪S(X_T∪S)−Eu_λ_T(X_T), S∈2^N\T.

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So, we can note that (λ_i₁Eu_i₁(Y_i₁), . . . , λ_i_tEu_i_t(Y_i_t))∈ M Cg (¯u) ∈ RM(¯u), with|T|=t,{i₁, . . . , it}=Tand, also, λj1Eui1(Yi1), . . . , λjn−tEujn−t(Yjt)

∈ M Cg (˜u)∈ RM(˜u) withN\T ={j₁, . . . , jn−t}because

X

i∈S⊂N\T

λiEui(Yi) = X

i∈S∪T

λiEui(Yi)−X

i∈T

λiEui(Yi)≥

≥EuλT∪S(XT∪S)−X

i∈T

λiEui(Yi) =

=Eu_λ_T∪S(X_T∪S)−Eu_λ_T(X_T) =Eu˜_λ_S(X_S), ∀S∈2^N\T and

X

i∈N\T

λ_iEu_i(Y_i) = X

i∈N

λ_iEu_i(Y_i)−X

i∈T

λ_iEu_i(Y_i) =

= Eu_λ_N(XN)−Eu_λ_T(XT) = Eu˜_λ_N\T X_N_\T . Since|T|< nthe induction hypothesis implies that

(λi1Eui1(Yi1), . . . , λitEuit(Yit))∈W (Π (N)) (¯u)∈ RM(¯u) and, also, for |N\T|< n

λ_j₁Eu_i₁(Y_i₁), . . . , λ_j_n−tEu_j_n−t(Y_j_t)

∈W(Π (N)) (˜u)∈ RM(˜u). Then, λ_i₁Eu_i₁(Y_i₁), . . . , λ_i_tEu_i_t(Y_i_t), λ_j₁Eu_i₁(Y_i₁), . . . , λ_j_n−tEu_j_n−t(Y_j_t)

∈ W (Π(N)) (¯u)×W (Π(N)) (˜u)⊂W (Π(N)).We can prove this last inclusion as follows:

The extreme points who belong toW(Π(N)) (¯u)×W(Π(N)) (˜u) can be given by (m^ρ, m^τ) whereρ :{1, . . . ,|T|} →T and τ :{1, . . . ,|N\T|} →N\T are two bijections and m^ρ∈R^|T^|, m^τ ∈R^|N\T^| with the next form

m^ρ_ρ(1):=λ_ρ(1)Eu¯_ρ(1) X_ρ(1)

=λ_ρ(1)Eu_ρ(1) X_ρ(1) , m^ρ_ρ(2):=Eu¯λ{ρ(1),ρ(2)}

Xλ{ρ(1),ρ(2)}

−λ_ρ(1)Eu¯_ρ(1) X_ρ(1)

=

=Eu_λ{ρ(1),ρ(2)}

X_λ{ρ(1),ρ(2)}

−λ_ρ(1)Eu_ρ(1) X_ρ(1) , ...

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m^ρ_ρ(|T_|):=Eu¯λT(X_T)−Eu¯λT\{ρ(|T|)}

X_T\{ρ(|T|)}

=

=Eu_λ_T (X_T)−Eu_λ_T\{ρ(|T|)}(X_T\{ρ(|T|)}), m^τ_τ(1):=λ_τ(1)Eu˜_τ(1) X_τ(1)

=λ_τ(1)Eu_τ(1) X_τ(1) , m^τ_τ(2):=Eu˜λ{τ(1),τ(2)}

Xλ{τ(1),τ(2)}

−λ_τ(1)Eu˜_τ(1) X_τ(1)

=

=Eu_λ{τ(1),τ(2)}

X_λ{τ(1),τ(2)}

−λ_τ(1)Eu_τ(1) X_τ(1) , ...

m^τ_τ(|N\T_|):=Eu˜_λ_N\T (X_T)−Eu˜_λN\T\{ρ(|N\T|)}

XN\T\{ρ(|N\T|)}

=

=Eu_λ_N\T X_N\T

−Eu_λN\T\{ρ(|N\T|)}

XN\T\{ρ(|N\T|)}

. The vector (m^ρ, m^τ)∈Rⁿ corresponds to the marginal vector m^σ inW where σ:N→Nis defined by

σ(i) :=

( ρ(i) if 1≤i≤ |T|, τ(i− |T|) if|T|+ 1≤i≤n.

So, we have shown that ext(W(Π(N)) (¯u)×W (Π(N)) (˜u)) ⊂ W. But W (Π(N)) is a convex set, so, W(Π(N)) (¯u)×W(Π(N)) (˜u) ⊂ W(Π(N)). We have concluded that (λ1Eu1(Y1), . . . , λnEun(Yn)) ∈ ext M Cg

implies (λ₁Eu₁(Y₁), . . . , λ_nEu_n(Y_n))∈W(Π(N)).

We have

M C =n

λ|(λ₁Eu₁(Y₁), . . . , λ_nEu_n(Y_n))∈M Cgo and

W ={λ|(λ₁Eu₁(Y₁), . . . , λ_nEu_n(Y_n))∈W(Π (N))}.

Remark 3.3. LethN,{Eu_λ_S(X_S)}_S⊆Nibe the reinsurance market. Then, M C ⊂W.

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Received 6 March 2012 Spiru Haret University

Faculty of Mathematics and Informatics Str. Ion Ghica 13

030045 Bucharest, Romania