Fairness principles for insurance contracts in the presence of default risk
aa r X i v : . [ q -f i n . M F ] S e p FAIRNESS PRINCIPLES FOR INSURANCE CONTRACTS IN THEPRESENCE OF DEFAULT RISK
DELIA COCULESCU AND FREDDY DELBAEN
Abstract.
We use the theory of cooperative games for the design of fair insurance con-tracts. An insurance contract needs to specify the premium to be paid and a possibleparticipation in the benefit (or surplus) of the company. It results from the analysis thatwhen a contract is exposed to the default risk of the insurance company, ex-ante equilibriumconsiderations require a certain participation in the benefit of the company to be specifiedin the contracts. The fair benefit participation of agents appears as an outcome of a gameinvolving the residual risks induced by the default possibility and using fuzzy coalitions. Introduction
In insurance theory in contrast to financial theory, we cannot use the equilibrium consider-ations of the arbitrage pricing theory for valuation purposes. Such a use would require thatagents are able to sell short risk exposures or to hold risky positions of their choice. Instead,in insurance, the principle of insurable interest holds and this imposes that agents enteringinsurance contracts have an exposure to the insured risks. The only available actions byagents remain decisions on accepting or not accepting the proposed insurance contracts. Inthis situation, cooperative game theory is the right tool, as it provides equilibrium conceptsthat can be used for valuation purposes. This paper is intended to exploit this idea by char-acterising fair contracts in presence of default risk. It is worth noting that similarly to thearbitrage theory, we will avoid dealing with utility functions of economic agents and onlyfocus on the relation between the premium paid and the payoffs of the contracts.The common approach in insurance is to use so-called premium calculation principles,having plausible properties from an economic standpoint. One such largely agreed uponproperty is convexity, which ensures that diversification of risks, which is desirable from aneconomic pointy of view, is accounted for in the premium calculation. Fair allocation of totalpremia to individual contracts using coherent risk measures and cooperative game theorywas already employed in the literature (see [11]); fair allocations being defined as allocationsin the core of a cooperative game. We shall use this type of framework as a starting point.But the approaches so far completely left out the default risk of the insurance companyfrom the analysis. Indeed, in presence of default risk, the insured agents may remain exposedto a residual risk even after signing an insurance contract, as they might not fully receivethe promised indemnities. Furthermore, the insured agents are not similarly exposed tothe default risk of the insurance company, their loss in default being determined by thedependence between individual risk and total risk. Thus, the question of the fairness of theinsurance contracts comes in naturally whenever the insurance company may default.
Date : September 10, 2020. e choose to treat the problem of default risk as a time 1 allocation, or allocation of con-tracts’ random payments, as opposed to the premia allocation, which is a time 0 allocation,or cost allocation. This time 1 allocation consists in characterising the payoffs that are tobe paid at time 1 for each contract, depending on the realisation of the risks and whetherthe company defaults or not. In our approach, agents can form coalitions and compare costsand residual risks of the corresponding defaultable contracts. The residual risk of one agentmeasures the impact of the default risk of the insurance firm on the corresponding contract.Problems related to allocations of random payoffs have a long history in insurance, forinstance in the study of the optimal reinsurance or optimal risk transfer. Already Borch[9] mentions cooperative game theory as a mean of selecting among Pareto equilibria ina reinsurance problem. More recently and using coherent functions (either utilities or riskmeasures) there is a vast literature on optimal risk transfer that treats also time 1 allocations.We refer to Heath and Ku [16], Barrieu and El Karoui [5], [6], Jouini et al. [18], Filipovi´cand Kupper [17], Burgert and R¨uschendorf [8]. Conceptually, the approach in the currentpaper is distinct from the literature on optimal risk transfers using coherent functions. Maindifferences are underlined in the title. First, we are interested in fair payoffs of the contracts,while the literature on risk transfers aims at characterising Pareto optima, given specifiedutilities for agents. We will introduce fairness at the same time as Pareto optimality. Insteadof defining utility functions for the agents, we use a predefined, coherent price system and thedesire of agents of paying the least of costs. Secondly, and more fundamentally, in the currentpaper the main question is the default risk of the insurance contracts. In the above mentionedliterature, agents hold risky positions and at time 1 in equilibrium, some agents pay otheragents (hence the name of risk transfer). This is not a classical situation in the context whereagents are the insured: in general, insured agents do not make ex-post monetary transfersto compensate for losses that other insured agents incurred. Consequently, we propose thatall payments at time 1 are made from and within the limits of the existing capital of theinsurance firm. As claims in default are determined by bankruptcy procedures, we introducethe class of admissible payments that fulfil bankruptcy priority rules.The remaining of this paper is organised as follows: in Section 2 we introduce the pre-cise mathematical setting of premium calculation principles that we are using, which arecommonotonic submodular functions. Section 3 presents the insurance model the economicproblem to solve. Section 4 formultates the fairness principles in a game theoretic framework.Solving the time 1 allocation requires additional theoretical development. This is introducedin Section 5 where we propose the concept of state coalitions as a mean of reducing the setof solutions. This allows a more precise characterisation of fair payoffs and the proofs of themain results follow immediately. 2. Setup and notation
We work in a simple model consisting of two dates: time 0, where everything is known, anda fixed future date, time 1, where randomness is involved. For the purpose of representing thepossible outcomes at time 1, a probability space (Ω , F , P ) is fixed. Unless otherwise specified,all equalities and inequalities involving random variables are to be considered in an P a.s.sense. The space of losses (or claims) occurring time 1 is considered to be L ∞ (Ω , F , P ),simply denoted L ∞ , i.e., the collection of all essentially bounded random variables. t time 0, the (manager of the) insurance company needs to evaluate the liabilities attime 1. The valuation method chosen will have an impact on the pricing of the individualinsurance contracts as it will be explained later on. We shall assume that the total liabilityin the company’s balance sheet is evaluated by means of a convex functional P : L ∞ → R fulfilling the properties detailed below. Definition 1.
A mapping P : L ∞ → R is called a convex valuation function if the followingproperties hold(1) if 0 ≤ ξ ∈ L ∞ then P ( ξ ) ≥ P is convex i.e. for all ξ, η ∈ L ∞ , 0 ≤ λ ≤ P ( λξ + (1 − λ ) η ) ≤ λ P ( ξ ) +(1 − λ ) P ( η ),(3) for a ∈ R and ξ ∈ L ∞ , P ( ξ + a ) = P ( ξ ) + a (4) if ξ n ↑ ξ (with ξ n ∈ L ∞ ) then P ( ξ n ) → P ( ξ ).If moreover for all 0 ≤ λ ∈ R , P ( λξ ) = λ P ( ξ ), we call P coherent.The number P ( ξ ) may be interpreted as a risk adjusted valuation of the future uncertainposition ξ . More specifically, we shall consider P to be a premium principle that the insurancecompany uses to determine the total premia to be collected from the insured, when theinsurance company faces liability ξ at time 1. This reflects the view that insurance premiashould be dependent on the whole portfolio of insurance contracts (see Deprez and Gerber[15]).Property (1) in the definition is therefore clear: liability tomorrow brings cash todayfor the insurance firm; or, conversely, premium collected time 0 corresponds to liabilityin insurer’s balance sheet. The convexity property is a translation of the diversificationbenefit. Combinations of risks are less risky than individual positions and, in presenceof high competition on the insurance market, it is reasonable to believe that the benefitsfrom the diversification will be passed along to the insured, at least partially, through lowerpremia. This explains property (2). Property (3) means that risk adjusted valuations aremeasured in money units. Of course money time 0 is different from money at the end of theperiod. Introducing a deflator or discounting – as is the practice in actuarial business sincehundreds of years – solves this problem. It complicates notation and as long as there is onlyone currency involved it does not lead to confusion if one supposes that this discounting isalready incorporated in the variables. The fourth property is a continuity property. Usingmonotonicity (a consequence of the previous properties, see [13]), we can also require that P ( ξ n ) ↑ P ( ξ ). The homogeneity property is a strong property. Remark . A coherent valuation function P is a submodular function. If we put ρ ( ξ ) = P ( − ξ ) we obtain a coherent risk measure (having the Fatou property); alternatively u ( ξ ) = −P ( − ξ ) defines a coherent utility function (see [12] for more details). Submodular functionsthat are commonotonic are commonly used in insurance as nonlinear premium principles.We will come back to this below, once we define commonotonicity.We say that a random variable ξ is acceptable if P ( ξ ) ≤
0. Remark that ξ − P ( ξ ) is alwaysacceptable. If P is coherent, then the acceptability set A := { ξ | P ( ξ ) ≤ } is a convex cone. In this paper, the term “acceptability” refers to the insurance companyand its balance sheet, and not to agents’ preferences. Acceptable risks have positive value for he insurance company to hold, hence it would accept holding these risks without requiringa premium payment in exchange.The continuity assumption allows to apply convex duality theory and leads to the followingrepresentation theorem Theorem 1. If P is coherent, there exists a convex closed set S ⊂ L (with L being thespace of all equivalence classes of integrable random variables on (Ω , F , P ) ), consisting ofprobability measures, absolutely continuous with respect to P , such that for all ξ ∈ L ∞ : P ( ξ ) = sup Q ∈S E Q [ ξ ] . (2.1) Conversely each such a set S defines a coherent valuation function. We will consider such a set S as given and fixed through the analysis, and we will referto it as the “scenario set” generating P . An assumption that we will make in this text isthat P is coherent with the set S being weakly compact. The weak compactness property of S is equivalent to a more stronger continuity property of P than in Definition 1 (4), where ↑ ’s are replaced by ↓ ’s. The weak compactness ensures that given ξ ∈ L ∞ one can find aprobability measure Q ξ ∈ S such that P ( ξ ) = E Q ξ [ ξ ] . Indeed, the weak compactness of S is also equivalent to the weak subgradient of P at ξ , ∇P ( ξ ), being non empty for all ξ ∈ L ∞ . We refer to [13] for the precise statements andproofs of equivalent formulations for S being weakly compact.In this text we will also make the assumption that P is commonotonic. Definition 2.
We say that two random variables ξ, η are commonotonic if there exist arandom variable ζ as well as two non-decreasing functions f, g : R → R such that ξ = f ( ζ )and η = g ( ζ ).It is easily seen that two random variables ξ, η such that ξ ≤ , η ≥ { ξ < } ∩ { η > } = ∅ , are always commonotonic. Definition 3.
We say that P : L ∞ → R is commonotonic if for each couple ξ, η of com-monotonic random variables we have P ( ξ + η ) = P ( ξ ) + P ( η ). Remark . Commonotonic convex monetary valuation functions are positively homogeneousand hence coherent.Addition of commonotonic risks is the opposite of diversification. Indeed, ξ and η beingnondecreasing functions of ζ , neither of them is a hedge against the other. The commono-tonicity of P can therefore be seen as a translation of the rule: if there is no diversification,there is also no gain when putting these claims together. Including commotonicity as aneconomic principle is considered to be desirable for the purpose of premium calculation orrisk measurement in insurance and many actuarial models are built on the assumption thatthe premium principles are commonotonic. For instance, the popular class of concave dis-tortion risk measures are commonotonic risk measures, hence they can be seen as particularexamples fitting in the framework we develop below.With P commonotonic, one can prove the following result (this will be useful later on): emma 1. Let ξ ∈ L ∞ and m ∈ R . Then, the following hold: ∇P ( ξ ) ⊂∇P (( ξ − m ) + ) (2.2) ∇P ( ξ ) ⊂∇P ( − ( ξ − m ) − ) (2.3) ∇P ( ξ ) ⊂∇P ( ξ ∧ m ) . (2.4) Proof.
Using commonotonicity of P we find P ( ξ − m ) = P (cid:0) ( ξ − m ) + (cid:1) + P (cid:0) − ( ξ − m ) − (cid:1) (2.5)Any Q ∗ ∈ ∇P ( ξ ) satisfies: P ( ξ − m ) = sup Q ∈S E Q [ ξ − m ] = E Q ∗ [ ξ − m ] P (cid:0) ( ξ − m ) + (cid:1) ≥ E Q ∗ (cid:0) ( ξ − m ) + (cid:1) P (cid:0) − ( ξ − m ) − (cid:1) ≥ E Q ∗ (cid:0) − ( ξ − m ) − (cid:1) then replacing these expressions in (2.5) we find that P ( ξ − m ) = P (cid:0) ( ξ − m ) + (cid:1) + P (cid:0) − ( ξ − m ) − (cid:1) (2.6) ≥ E Q ∗ [ ξ − m ] = P ( ξ − m ) , (2.7)in other words we only have equalities. This proves that Q ∗ ∈ ∇P (( ξ − m ) + ) and Q ∗ ∈∇P ( − ( ξ − m ) − ). Similarily, by commonotonicity and the above: P ( ξ ∧ m ) = P ( ξ ) − P (cid:0) ( ξ − m ) + (cid:1) = E Q ∗ [ ξ ∧ m ] , proving Q ∗ ∈ ∇P ( ξ ∧ m ). (cid:3) There is a link between commonotonic valuation functions and Choquet integration theory(see Schmeidler [20]). For bounded nonnegative risks ξ , the following representation holdswhen P is a commonotonic valuation function: P ( ξ ) = sup µ ∈C ( w ) E µ [ ξ ] = Z ∞ w ( ξ > a ) da (2.8)where w : F → R + satisfies w (Ω) = 1 and w ( A ∩ B ) + w ( A ∪ B ) ≤ w ( A ) + w ( B ) , (2.9)and C ( w ) := { µ finitely additive measure | µ (Ω) = 1 , ∀ A ∈ F : 0 ≤ µ ( A ) ≤ w ( A ) = P ( A ) } . The use of Choquet integration as premium principle was emphasized by Denneberg, [14].Denneberg was inspired by the pioneering work of Yaari, [22].A set function w satisfying (2.9) is called 2-alternating or supermodular and can serve asa characteristic function for cost games (which are the duals of convex games) and C ( w ) iscalled the core of the game. In general S ⊂ C ( w ), but when S is weakly compact we have S = C ( w ) and the following representation of the scenario set S holds: S = { µ probability measure | µ ≪ P , µ ( A ) ≤ w ( A ) , ∀ A ∈ F } . In Delbaen [10] and Schmeidler [19] one can find the basics of convex game theory that aregoing to be used in this paper. o sum up, the properties of P that are going to be used in this paper are: Assumptions.
The valuation function P is commonotonic and the scenario set S (cf. therepresentation in (2.1)) is weakly compact in L .3. The economic model
We consider N economic agents ( N ∈ N ∗ ) that are the potential buyers of insurance.Each agent i is endowed with a risky position X i ∈ L ∞ + and is considering at time 0 theoption to buy insurance for covering his/her exposure. The insurer will be denoted by theindex 0; his role is to propose insurance contracts to each agent. Also, he provides an initialcapital denoted by k . We shall propose some principles for the design of fair individualinsurance contracts. The only assumption about the agents preferences is that they prefermore rather than less and they are risk adverse, as they seek to buy insurance. Therefore,when comparing different contracts that are proposed to them, the premium to be paid isan important element in their decision making. They choose the lower premium when theindemnities are the same. We will have to deal with the issue that the default risk of theinsurance company is depending on the premia collected, impacting the cash flows of thecontracts at time 1. Therefore random cash flows need to be compared as well and notonly premia. This will be detailed in a while (Section 4). We shall assume the insurer’scontribution k exogenous to the analysis, for instance it reflects some exogenous regulatoryconstraints and shareholder preferences regarding the riskiness of the company’s shares. Thequestion of setting a level of the capital k is not addressed in this paper.Therefore, we fix X = ( X , X , ..., X N ), i.e. an N + 1 dimensional vector of randomvariables nonnegative and bounded, where X , ..., X N are the risks to be insured, while X = k by convention, i.e, the equity capital brought by the insurer at time 0. For simplicity,from now on we shall refer to X as the liability vector , even though, the first component X is strictly speaking not liability. From now on, N + 1 dimensional random vectors will alwaysappear in bold.The insurance company uses a commonotonic valuation function P for calculating thetotal premia to be collected. Given that the total risk exposure is S X := N X i =1 X i , the total premium is computed as: k := P (cid:0) S X (cid:1) (3.1)so that the total capital of the insurance firm is K := k + k , and we will suppose that suppose that P ( S X > k ) > K >
0. These assumptions exclude sometrivial cases from the analysis.Hence, we assume the insurance company charges the insured the minimum amount thatmakes the aggregate net position acceptable. This reflects the discussion from the previoussection, where we assumed that there are benefits from pooling risks and these benefits are We use this term in the frame of this paper to mean total fund that is available in the insurance firm attime 1; it does not match an accounting definition of total capital. assed along to the insured, at least partly. The valuation of the total premia using a convexfunctional as in (3.1) is a necessary first step in order to pass along some diversificationbenefits to the insured. Remains to ensure that each agent individually “gets her share ofthe pie”; here the concept of fairness of the contracts comes into play.Let us more precisely describe the components of an insurance contract.3.1. Premia, indemnities and benefit sharing.
The insurance company is proposinginsurance contracts to the N agents. In particular, for the risks X , · · · X N and with aninitial capital X = k , the insurance company proposes contracts(( π , X , B ) , · · · , ( π N , X N , B N )) ∈ ( R + × L ∞ + × L ∞ + ) N i.e., a collection of individual contracts, with each individual contract ( π i , X i , B i ) specifyinga premium π i ≥ i at time 0 and a promised payoff of the contractat time 1, consisting of the indemnity X i and possibly a participation in the benefit of thecompany, B i ≥ B i = 0, i.e., there is no suchbenefit sharing proposed in the contracts. In this case, the insurer will receive the wholebenefit in the form of a dividend. We shall deal with the issue of the benefit sharing usingequilibrium considerations, rather than taking from the outset the point of view that thereis or not such a participation proposed within a contract.The actual payments cannot exceed the total capital of the insurance company, and hencethe insurance company defaults when S X > K . The contracts are defaultable and thereforethe promised payment X i + B i may differ from the actual payment of the contract, denoted Y i . This satisfies Y i = X i + B i if the company does not default Y i < X i if the company defaults on its contracts.It follows that the actual payment and the benefit participation are linked by the relation B i = ( Y i − X i ) + . As the insurer brings in an initial capital k , he is entitled at time 1 to a dividend Y := K − N X i =1 Y i ! + , that is, the payoff to the insurer is the residual value, once all payoffs to the insured are paidout.All payments by the insurance company occurring at time 1 will be called payoffs ; theyare indemnities and dividends and they represent the actual monetary transfers from theinsurance firm toward the agents, as opposed to the promised payments of the contracts.We shall restrict our analysis to the class of payoffs that are feasible, i.e., they do not exceedthe total capital available and also respect the legal requirement that indemnities have ahigher priority of payment over dividends. These are called admissible payoffs. Definition 4.
We denote by X the space of N + 1 dimensional random variables which arebounded. We consider the liability vector X = ( X , ..., X N ) ∈ X with all X i ≥ X constant and consider K another constant satisfying X ≤ K . a) We call payoff of total mass K any vector Y = ( Y , · · · , Y N ) ∈ X that satisfies P Ni =0 Y i = K and each Y i is σ ( X i , S X ) measurable. The class of payoffs of total mass K is denoted by X ( X , K ).(b) The class of admissible payoffs of total mass K , corresponding to the liability X =( X , ..., X N ) ∈ X is defined as: A X ( K ) = (cid:26) Y ∈ X ( X , K ) (cid:12)(cid:12)(cid:12)(cid:12) ∀ i ≥ Y i { S X >K } = X i S X K { S X >K } Y i { S X ≤ K } ≥ X i { S X ≤ K } (cid:27) . Admissible payoffs respect the following rules: all insurance claims X i , i ≥ S X ≤ K ); default occurswhen the total insurance claim exceeds the total capital ( S X > K ); in case of default nodividend is distributed and the insured have equal priority of their claims. Hence in defaultall capital K is distributed as indemnities, proportionally to the claim size. The fact thatan admissible payoff component Y i is considered σ ( X i , S X ) measurable ensures that we canspecify ex-ante within the contracts the form of the payoffs as functions of the individualrisk and total risk only. This ensures that each agent can observe time 1 their payoff andjudge if it respects the contract, without necessarily having knowledge of the losses of theother agents individually. This assumption reflects the practice in the industry. From amathematical point of view it is not a necessary condition though. Example . Standard payoffs . Let us consider that each insured receives some constantproportion of the company’s benefit (cid:0) k − S X (cid:1) + . In this case, these constant proportions canbe specified in the contracts at time 0. The corresponding admissible payoffs (that we shallcall standard payoffs) are given as follows: Y i = (cid:2) X i + α i (cid:0) k − S X (cid:1)(cid:3) { S X ≤ k } + X i (cid:18) KS X ∧ (cid:19) { S X >k } , i = 1 , . . . , N (3.2) Y = (cid:2) k + α (cid:0) k − S X (cid:1)(cid:3) { S X ≤ k } + (cid:0) K − S X (cid:1) + { S X >k } , (3.3)where k = K − X = K − k is the total premium and each α i is a nonnegative constantand P Ni =0 α i = 1. The standard allocations have the feature that the surplus (or benefit)( K − S X ) + is shared between the agents and the insurer as follows:- when S X ∈ [ k, K ], the insurer takes all the surplus, as it does not exceed his initialcontribution (the equity capital) k .- when S X < k , there is a benefit and all agents and the insurer receive some fixedshare of the benefit.For the insured the benefit writes: B i = α i (cid:0) k − S X (cid:1) + . Definition 5.
The payoff vector Y is called standard if they are of the form (3.2)-(3.3). Thecorresponding contracts ( π i , Y i ) are then called standard .3.2. Problem description.
A liability vector X ∈ X is fixed, with X = k ≥ X i requires specifying two components: π i ,the premium and B i ≥
0, i.e., a possible benefit participation. This is the same as specifyinga premium π i and a random payment Y i , the actual contract payment. We saw above that he actual payment Y i and the the benefit participation are linked by: B i = ( Y i − X i ) { S X In this section we define the fair contracts as outcomes of cooperative games. The notionof a fair cost allocation that is based on equilibrium in a cooperative game, was introducedalready in Delbaen [11], [13]. We go further and also employ convex games for solving thetime 1 allocation problem, that is, the determination of the family of random payoffs Y .In [11] it was shown that starting from any given risk vector X = ( X , ..., X N ) and givena coherent valuation function P , one can define a cooperative game as follows. Let N := { , ..., N } be the set of players, consisting of all agents and the insurer; 2 N is the set of allpossible groups of players, named coalitions. We refer to N as the grand coalition. We thendefine a cost game ( N , N , c X ), with c X being the characteristic cost function of the game, c X : 2 N → R defined as: c X ( S ) := P X i ∈ S X i ! , for all S ∈ N . In the literature, one often encounters a value function of the game instead of a cost function.The value function v X is obtained as v X ( S ) := c X ( N ) − c X ( N \ S ); v X expresses value (orprofit) and eplayers aim to get the most possible, while c X measures costs that players intendto minimise. As we study cost allocation here, it is more intuitive to use the cost function c X , instead of the value function v X . All concepts from profit games can easily be translatedto the cost setting. he set of actions available to a coalition S consists of all possible divisions ( x i ) , i ∈ S of c X ( S ) among the members of S , P i ∈ S x i = c X ( S ). We search for an action of the grandcoalition that is stable, in the sense that no coalition S can obtain a better outcome. Theset of such actions is called the core of the game (see Shapley [21] for more details): C (cid:0) c X (cid:1) := ( x = ( x i , i ∈ N ) (cid:12)(cid:12)(cid:12) X i ∈N x i = c X ( N ) and for all S ∈ N X i ∈ S x i ≤ c X ( S ) ) We follow [11] and propose the following definition: Definition 6. An element π = ( π i , i ∈ N ) ∈ C ( c X ) is called a fair cost allocation for therisk vector X .We remark that if π ∈ C (cid:0) c X (cid:1) we necessarily have that π = k . Indeed π ≤ c X ( k ) = k and P j ≥ π j ≤ P ( P j ≥ X j ) = k . But P j ≥ π j = K = k + k and hence the two inequalitiesmust be equalities.The characteristic function c X gives for any coalition the value of its liability, using thevaluation function of the insurance company P . From the standpoint of the agents in acoalition S , c X ( S ) represents a cost they have to bear for insurance, should they decide tosplit from the grand coalition. A fair premia is therefore simply any repartition of the totalcapital K in N + 1 individual costs representing the contribution of each agent, such thatno coalition prefers to split from the grand coalition in order to become a separate entity tobe insured.Once the costs allocated, the next problem is determining the payoffs of the contracts, inparticular a fair benefit allocation. This requires to specify a preference order on the spaceof payoff vectors X , i.e., the space of bounded, N + 1 dimensional random variables. As saidpreviously, we do not assume any specific utility function for the agents, but simply thatthey are preferring to pay less rather than more and are risk adverse. In other words, theyhave increasing utility functions and care about the cost of insurance.The following definition introduces the most natural preference order for each coalition,namely the cheapest to insure payoff is preferred by any group of agents. Given that anyagent i has already an exposure X i , this gives the following: Definition 7. Given two payoff vectors ξ , η ∈ X , we say that the coalition S ∈ N prefers ξ to η , and we write η ≺ S ξ if: P X i ∈ S ( X i − ξ i ) ! < P X i ∈ S ( X i − η i ) ! . or, alternatively: c X − ξ ( S ) < c X − η ( S ) . The weak preference relation is denoted by (cid:22) S , namely we write η (cid:22) S ξ if the coalition S does not prefer η to ξ ; indifference is denoted by ∼ S . Remark . We notice that with our definition of preferences, a coalition S aggregates valuefrom each of its members and does not care about allocations of costs or payoffs among itsmembers: whenever ξ , η ∈ X satisfy P i ∈ S ξ i = P i ∈ S η i , the coalition is indifferent betweenthe two vectors: ξ ∼ S η . his is nothing but the concept of transferable utility, a common assumption in cooperativegames.Below we are going to apply preferences to net payoffs, which are payoffs from insuranceminus costs of insurance. It is therefore important to notice that a coalition S will beindifferent between = (0 , ..., 0) and any payoff ξ satisfying P i ∈ S ξ i = 0. Remark . Any risk ξ ∈ X satisfying P i ∈ S ( X i − ξ i ) ∈ A satisfies (cid:22) S ξ .We propose the following Definition 8. Given a fair cost allocation π , a fair payoff for the risk vector X , is a vectorof random variables Y ∈ A X ( K ) satisfying (cid:22) S Y − π ∀ S ∈ N , (4.1)with = (0 , ..., ∈ X .As a member of the grand coalition, player i ∈ N pays π i and receives at time 1 thepayoff Y i . Therefore Y − π represents the vector of net payoffs of players. A coalition S has therefore a net payoff P i ∈ S ( Y i − π i ). Alternatively, a coalition S has a total cost c X ( S )to be paid at time 0 and a total payoff of c X ( S ) that can be distributed to members of thecoalition at time 1. The net payoff at the coalition level is therefore 0. The interpretation of(4.1) is therefore the following: fair payoffs are such that any coalition S prefers Y − π to anull net cash flow, that is, prefers buying insurance within the grand coalition rather thandoing this as a separate entity.An alternative — maybe easier to understand — expression for (4.1) is the following c X − Y ( S ) + X i ∈ S π i ≤ c X ( S ) ∀ S ∈ N , (4.2)where c X − Y is the value of the residual claims of the coalition S , c X − Y ( S ) = P X i ∈ S ( X i − Y i ) ! . The intuition behind the expression (4.2) is as follows. As usual, player i is exposed to risk X i ; we consider each player aims at achieving full coverage of their risk at the lowest possiblecost. To achieve full coverage of risks, agents consider also the cost to insure residual risks P ( X i − Y i ), they need to pay additional insurance cost if this quantity is positive. Everycoalition S looks at the proposed cost allocation π and payoff allocation Y within the grandcoalition and compare with the stand alone situation.Hence, when the payoff Y i is distributed to player i as a member of the grand coalition,it leaves agent i with a residual risk X i − Y i . All players now need to cover their residualrisks. They can decide to form coalitions for this purpose. The residual cost of achievingan acceptable position for a coalition S is P (cid:0)P i ∈ S ( X i − Y i ) (cid:1) = c X − Y ( S ). Hence, all playersmay consider entering a second game (the re-insurance game), with characteristic function c X − Y .Alternatively, a coalition S , by paying a premia c X ( S ) achieves directly an acceptableresidual value and reinsurance is costless. Indeed, X i ∈ S X i − c X ( S ) ∈ A . f the cost allocation and the payoffs of the initial game ( N , N , c X ) are fair, any coalitioncan achieve an acceptable position at a lower cost as members of the grand coalition and withpossibly a reinsurance of the residual risk, rather than forming from the outset a separategroup.It is important to emphasize that in general the residual risk X i − Y i of an agent is nonzero, because of the default risk of the company.For this reason, in addition to fairness, it seems important that payoffs are tailored in away that achieves Pareto optimality, in the cost minimisation problem of the residual risksof all agents: Definition 9. A payoff family Y ∈ A X ( K ) is said to be maximal if it is a solution of thefollowing cost minimisation problem:inf ξ ∈ A X ( K ) N X i =0 P ( X i − ξ i ) . Lemma 2. Standard payoffs are maximal.Proof. For any ξ , η ∈ A X ( K ) we have N X i =0 P (( X i − ξ i ) + ) = N X i =0 P (( X i − η i ) + ) . (4.3)Indeed, we have that the acceptability of ξ implies for all i ∈ N : { X i − ξ i > } ⊂ { S X > k } and the payoffs in default are entirely fixed by the admissibility conditions: ξ i { S X >k } = η i { S X >k } . It follows that ( X i − ξ i ) + = ( X i − η i ) + , ∀ i ∈ N , hence (4.3) is verified.For any random variable ξ , P ( X i − ξ ) = P (( X i − ξ ) + ) + P ( − ( X i − ξ ) − ) by commono-tonicity of P . Using this property and (4.3), it follows that Y is maximal if and only if it isa solution of inf ξ ∈ A X ( K ) N X i =0 P (cid:0) − ( X i − ξ i ) − (cid:1) . (4.4)As P is convex, for any ξ ∈ A X ( K ): P Ni =0 P ( − ( X i − ξ i ) − ) ≥ P (cid:16) − P Ni =0 ( X i − ξ i ) − (cid:17) = P (cid:0) − ( S X − K ) − (cid:1) . So, the quantity in (4.4) is bounded below by P (cid:0) − ( S X − K ) − (cid:1) .To conclude, we notice that Y standard, implies all ( X i − Y i ) − commonotonic, so that P Ni =0 P ( − ( X i − Y i ) − ) = P (cid:16) − P Ni =0 ( X i − Y i ) − (cid:17) = P (cid:0) − ( S X − K ) − (cid:1) . (cid:3) We now return to the insurance problem and give some definitions for this framework. Definition 10. Consider the risks ( X i ) Ni =1 and some corresponding contracts { ( π i , Y i ) } Ni =1 with initial capital X = k . We denote π := K − P Ni =1 π i and Y = K − P Ni =1 Y i .(1) The premia ( π i ) Ni =1 are said to be fair if π = k and π = ( π , ..., π N ) is a fair costallocation for the risk vector X = ( X , ..., X N ).(2) The contracts are said to be fair if the premia ( π i ) Ni =1 are fair and, given π =( π , · · · , π N ), Y = ( Y , · · · , Y N ) is a fair payoff. . Determining fair contracts The aim of this section is to characterise some fair insurance contracts in presence ofdefault risk. The main difficulty is the design of fair payoffs for the defaultable contracts.As a technique to tackle this problem, we propose in Subsection 5.1 a distinct cooperativegame, where players are states of Ω and coalitions are elements of F . The interpretation isthat states are competing to get the highest possible payoff under the threat to split from thegrand coalition Ω. Another interpretation is that for each element A ∈ F , one can create anavailable contract that can be entered at time 0 to cover the risk A . That is, Arrow-Debreurisks can be covered by insurance contracts and an agent can enter such a contract andeven several such contracts, provided it does not exceed the agent’s true risk exposure (bythe principle of insurable interest). We will show that the fuzzy version of this game is infact a generalisation of the game that we introduced in the previous section. In Subsection5.2, we will prove that ensuring fairness in the “Arrow-Debreu fuzzy game” leads to fairnessof the contracts in the frame of the initial N + 1 player game.5.1. Arrow-Debreu claims and the families of fair state payoffs. We denote Z := S X + k . Let us consider the following cost game (Ω , F , c Z ) with the characteristic function c Z : F → R is defined as c Z ( A ) := P ( Z A ) , A ∈ F . Any element in F is now interpreted as a coalition, and F as the set of all coalitions. Thecost c Z ( A ) = P ( Z A ) corresponds to the liability level of the insurance company, relative tothe set A . By the principle of insurable interest, a coalition could not exceed this level ofexposure, that is the cumulative exposure of all agents. But lower levels should be possibleto reach by coalitions but they are currently excluded in this setting. For this reason, weintroduce fuzzy coalitions, where players ω ∈ Ω “choose” a rate of participation in a coalitioninstead of a binary decision yes/no: Definition 11. A fuzzy coalition is a random variable λ : (Ω , F ) → [0 , λ , the corresponding cost function is c Z : [0 , → R given as c Z ( λ ) := P ( Zλ ).From an economical standpoint, an insurance company that commits to liability Z im-plicitly commits to any lower liability 0 ≤ ξ ≤ Z as well. This level of risk corresponds toa fuzzy coalition λ = ξZ . The fuzzy game (Ω , F , c Z ) is a generalisation of the finite-playergame ( N , N , c X ) from the previous section. Indeed, the set of players N = { , ..., N } canbe linked to the following set of fuzzy coalitions { ˆ λ , ..., ˆ λ N } with ˆ λ i = X i Z so that the costfunction of the finite-player game is obtained via the relation c X ( S ) = c Z X i ∈ S ˆ λ i ! , for any S ∈ N . Such a market could be labelled as “complete”; however this could lead to some confusions, as we donot assume that individual agents can eliminate all risks, the contracts that they enter being defaultable.Also, note that we do not assume that for an A ∈ F there is necessarily an agent i ∈ { , .., N } having a riskexposure of A . roposition 1. The set function c Z is 2-alternating, that is, it satisfies for all A, B ∈ F : c Z ( A ∩ B ) + c Z ( A ∪ B ) ≤ c Z ( A ) + c Z ( B ) . This implies that the cost game (Ω , F , c Z ) is superadditive.Proof. For any ξ , ξ ∈ L ∞ + , it holds that c Z ( ξ ∨ ξ ) + c Z ( ξ ∧ ξ ) ≤ c Z ( ξ ) + c Z ( ξ ). Indeed,using the representation of P in (2.8)-(2.9): c Z ( ξ ∨ ξ ) + c Z ( ξ ∧ ξ ) = P ( Z ( ξ ∨ ξ )) + P ( Z ( ξ ∧ ξ ))= P (( Zξ ) ∨ ( Zξ )) + P (( Zξ ) ∧ ( Zξ ))= Z ∞ w (( Zξ ) ∨ ( Zξ ) > a ) da + Z ∞ w (( Zξ ) ∧ ( Zξ ) > a ) da = Z ∞ w ( { Zξ > a } ∪ { Zξ > a } ) da + Z ∞ w ( { Zξ > a } ∩ { Zξ > a } ) da ≤ Z ∞ w ( Zξ > a ) da + Z ∞ w ( Zξ > a ) da ≤ P ( Zξ ) + P ( Zξ ) = c Z ( ξ ) + c Z ( ξ ) . By taking ξ = A and ξ = B , we find the claimed inequality. (cid:3) In this framework, we consider a family: { ( π ( A ) , Y ( A )) , A ∈ F } , (5.1)where for every set A ∈ F , π ( A ) ∈ R + is interpreted as a cost and Y ( A ) is a randomvariable interpreted as a payoff, both corresponding to the risk exposure Z A . It remainsto define the notions of fair cost and fair payoffs in this setting. The definitions below are ageneralisation of the ones with finite number of coalitions 2 N +1 in Section 4. Definition 12. (i) A fair cost allocation is any element π in the core C ( c Z ) of the game, C ( c Z ) = (cid:8) ν ∈ ba (Ω , F , P ) | ν (Ω) = K = c Z (Ω) and ν ( A ) ≤ c Z ( A ) , ∀ A ∈ F (cid:9) . (ii) The fuzzy core e C ( c Z ) of the game is e C ( c Z ) = (cid:26) ν ∈ ba (Ω , F , P ) | ν (Ω) = K = c Z (Ω) and Z λdν ≤ P ( λZ ) , ∀ λ : (Ω , F ) → [0 , (cid:27) . A fair cost allocation in the sense of fuzzy games is any element π in the fuzzy coreof the game.The notation ba (Ω , F , P ) above stands for the space of bounded, finitely additive measuresabsolutely continuous with respect to P . It is easily seen that e C ( c Z ) ⊂ C ( c Z ). Also the weakcompactness of S implies that for A n ↓ ∅ , we have c Z ( A n ) ↓ 0. The core C ( c Z ) is thereforea set of sigma-additive measures, in other words e C ( c Z ) ⊂ C ( c Z ) ⊂ L and the two cores areweakly compact convex sets.The core is the set of allocations which cannot be improved upon by any coalition. Wecan “shrink” it by introducing the fuzzy core, that is the set of allocations which cannot beimproved upon by any fuzzy coalition. The basic papers for this approach are Aubin [4],Artzner and Ostroy [3] and Billera and Heath [7]. As explained above, using the fuzzy gameapproach is interesting as it leads to a generalisation of the game in the previous section. nother advantage of using the fuzzy game approach is that the fuzzy core can be char-acterised and linked to some elements in the scenario set S : Proposition 2. The following hold: e C (cid:0) c Z (cid:1) = Z · ∇P ( Z ) ⊂ C (cid:0) c Z (cid:1) Z · S ⊂ C (cid:0) c Z (cid:1) − L If Z > a.s. then e C (cid:0) c Z (cid:1) = C (cid:0) c Z (cid:1) implies P ( Z ) = max Q ∈S E Q [ Z ] = min Q ∈S E Q [ Z ] . Proof. We first show e C (cid:0) c Z (cid:1) = Z · ∇P ( Z ). By definition ∇P ( Z ) = S ∩ { ν | ν (Ω) = K } . Takenow ν ∈ Z · S , then obviously ν ( ξ ) ≤ c Z ( ξ ) for all ξ ∈ L ∞ . This shows that Z · ∇P ( Z ) ⊂ e C (cid:0) c Z (cid:1) . Conversely, take ν ∈ e C (cid:0) c Z (cid:1) then ν satisfies ν ( λ ) ≤ P ( λZ ) for all λ : (Ω , F ) → [0 , ν satisfies ν ( ξ ) ≤ P ( ξZ ) for all ξ ∈ L ∞ . Hence ν ∈ Z · S . But then ν (Ω) = K shows that ν ∈ Z · ∇P ( Z ).Let us now prove that Z · S ⊂ C (cid:0) c Z (cid:1) − L . Take Q ∈ S and suppose that Q / ∈ C (cid:0) c Z (cid:1) − L .The set C (cid:0) c Z (cid:1) is weakly compact and hence C (cid:0) c Z (cid:1) − L is a closed convex subset of L .The Hahn-Banach theorem then allows to find ξ ∈ L ∞ such that E Q [ ξZ ] > sup (cid:26) E (cid:20) ξ (cid:18) dνd P − h (cid:19)(cid:21) | ν ∈ C (cid:0) c Z (cid:1) ; h ∈ L (cid:27) . This implies that ξ ≥ { E [ ξ ( dν/d P − h )) | ν ∈ C (cid:0) c Z (cid:1) ; h ∈ L } = sup { ν ( ξ ) | ν ∈ C (cid:0) c Z (cid:1) } . We get that Z ∞ E Q [ Z { ξ>u } ] du = E Q [ ξZ ] > sup { ν ( ξ ) | ν ∈ C (cid:0) c Z (cid:1) } = Z ∞ c Z ( ξ > u ) du, which is a contradiction since E Q [ Z { ξ>u } ] ≤ c Z ( ξ > u ) for all u and all Q ∈ S . The inclusioncan be read as follows. For all Q ∈ S there exists ν ∈ C (cid:0) c Z (cid:1) so that Z · d Q d P ≤ dνd P .If e C (cid:0) c Z (cid:1) = C (cid:0) c Z (cid:1) , then, for all Q ∈ S there exists Q ∈ ∇P ( Z ) so that Z · d Q d P ≤ Z · d Q d P .If Z > a.s. then d Q d P ≤ d Q d P a.s. , that is, Q = Q and hence S = ∇P ( Z ). (cid:3) Remark . The above proposition shows that the ∧ , ∨ inequality in the beginning of theproof of Proposition 1 is not equivalent to commonotonicity. Indeed, the functional c Z is notcommonotonic since it is different from sup { ν ( ξ ) | ν ∈ C (cid:0) c Z (cid:1) } which is the uniquely definedcommonotonic extension of c Z restricted to F .We now define random payoffs { Y ( A ) , A ∈ F } as introduced in (5.1) and study theirfairness. Definition 13. A family of random variables { Y ( A ) , A ∈ F } is called a family of statepayoffs if Y is a finitely additive vector measure on Ω Y : F → L ∞ + A Y ( A ) , that satisfies:(1) Y (Ω) = K a.s. . 2) For each A ∈ F , Y ( A ) is a random variable on (Ω , σ ( Z, A )). Y ( A ) shall be referred to as payoff for the coalition A .We make the choice that payoffs { Y ( A ) , A ∈ F } are measurable with respect to σ ( Z, A )rather than F . This is in order to have contracts that specify payoffs contingent on the riskrealisation and do not include some other extraneous randomness. Here also, this condition isnot needed from a mathematical point of view, but it is more realistic from an economic pointof view. In the definition we only required that Y is finitely additive. But the interestingpayoffs will become countably additive in the following sense. If ( A n ) n is a sequence ofpairwise disjoint sets taken in F , then Y ( A ) + Y ( A ) + ... = Y ( ∪ n A n ) = K. where the sum converges in probability (and not necessarily in L ∞ norm).We now define the admissible payoffs as those state payoffs that satisfy the followingconstraints : Definition 14. A family of state payoffs Y is admissible if Y ∈ A Z, F ( K ), where: A Z, F ( K ) = (cid:26) Y state payoff (cid:12)(cid:12)(cid:12)(cid:12) A ⊂ { Z ≥ K } ⇒ Y ( A ) A = K A A ⊂ { Z < K } ⇒ Y ( A ) ≥ Z A (cid:27) . Lemma 3. Any state payoff Y = { Y ( A ) , A ∈ F } ∈ A Z, F ( K ) has the representation: Y ( A ) = α ( A )( K − Z ) + + ( Z ∧ K ) A , (5.2) where α ( A ) is an F measurable random variable with values in [0 , . The mapping α : F → L ∞ + is a finitely additive vector measure, normalised to 1, i.e. α (Ω) = 1 a.s. . We will referto α as the benefit sharing measure corresponding to Y .Proof. Take Y ∈ A Z, F ( K ). We claim that if A ⊂ { Z ≥ K } then Y ( A ) { Z ≥ K } = K A and if B ⊂ { Z < K } then Y ( B ) { Z ≥ K } = 0. Indeed, if A ⊂ { Z ≥ K } , denoting e A := { Z ≥ K } \ A and using the additivity, we observe that Y ( { Z ≥ K } ) { Z ≥ K } = K { Z ≥ K } = (cid:16) Y ( A ) + Y ( e A ) (cid:17) { Z ≥ K } . As Y ( A ) A = K A and Y ( e A ) e A = K e A by the definition ofadmissible payoffs, it follows that Y ( A ) e A = 0. If B ⊂ { Z < K } then we denote e B := { Z ≥ K } ∪ B and we obtain Y ( e B ) { Z ≥ K } = ( Y ( B ) + Y ( { Z ≥ K } )) { Z ≥ K } ≤ K { Z ≥ K } .As Y ( { Z ≥ K } ) { Z ≥ K } = K { Z ≥ K } and Y is nonnegative, it follows that Y ( B ) { Z ≥ K } = 0.Hence both claims are proved. We obtain for A ⊂ { Z ≥ K } and B ⊂ { Z < K } therepresentations: Y ( A ) = K A + Y ( A ) { Z Definition 16. A family of state payoffs Y = { Y ( A ) , A ∈ F } is called maximal if for anyfinite sequence of mutually disjoint sets ( A n ) satisfying A n ∈ F and ∪ A n = Ω we have that { Y ( A n ) } n ≥ is a solution of the following cost minimisation problem:inf Y ′ ∈ A Z, F ( K ) X n P ( Z A n − Y ′ ( A n )) . (5.5) The reader who is not familiar with integration with respect to a finitely additive vector measure caneasily find out that the definition for elementary functions λ can be extended by using the L ∞ density. aximal payoffs are such that the costs of residual risks cannot be reduced even byreallocations of the risks in different portfolios (or coalitions), where a reallocation of riskscorresponds to a partition ( A n ) of Ω.The following result gathers some properties of maximal payoffs: Theorem 2. (a) Let α : (Ω , F ) → [0 , be a finitely additive probability measure. Then,the family of state payoffs Y associated to α via the relations (5.2) is maximal. (b) If e Y is maximal, there is a finitely additive probability measure α : (Ω , F ) → [0 , sothat Y ( A ) ∼ A e Y ( A ) , ∀ A ∈ F , where the payoff Y is associated to α via (5.2). (c) If e Y is maximal, then for Q ∗ ∈ ∇P ( Z ) : P (cid:18) − (cid:16) Z A − e Y ( A ) (cid:17) − (cid:19) = − E Q ∗ (cid:20)(cid:16) Z A − e Y ( A ) (cid:17) − (cid:21) , ∀ A ∈ F Proof. (a) By Lemma 3, a state payoff e Y = { e Y ( A ) , A ∈ F } ∈ A Z, F ( K ) writes: e Y ( A ) = e α ( A )( K − Z ) + + ( Z ∧ K ) A , where e α ( A ) is a vector measure, i.e., the benefit sharing measure. Using commono-tonicity of P we obtain: P ( Z A − e Y ( A )) = P (cid:0) − e α ( A )( K − Z ) + + ( Z − K ) + A (cid:1) = P (cid:0) − e α ( A )( K − Z ) + (cid:1) + P (cid:0) ( Z − K ) + A (cid:1) and therefore the infimum in (5.5) is obtained by a family of state payoffs Y with itscorresponding benefit sharing measure α being a solution ofinf e α X k P (cid:0) − e α ( A k )( K − Z ) + (cid:1) , (5.6)(the infimum being taken over the class of all vector measures on F ), and this forany ( A k ), finite partition of Ω. We observe that X k P (cid:0) − e α ( A k )( K − Z ) + (cid:1) ≥ P − X k e α ( A k )( K − Z ) + ! = P (cid:0) − ( K − Z ) + (cid:1) . In case ( e α ( A ) , e α ( A ) , · · · , e α ( A n )) is non random, we can use homogeneity of P toshow that we get equality above instead of an inequality, so that the infimum isattained by this vector. Therefore any finitely additive probability measure α on F is a solution of (5.6). Furthermore the infimum in (5.5) equals P (cid:0) − ( K − Z ) + (cid:1) + X k P (cid:0) ( Z − K ) + A k (cid:1) . (b) In general, for a solution e α of (5.6), we denote α ( A ) := P ( − e α ( A )( K − Z ) + ) P ( − ( K − Z ) + ) , ∀ A ∈ F . hen, α is a finitely additive measure. Indeed it is nonnegative, α (Ω) = 1 and finiteadditivity is also verified: for any ( A k ), a finite partition of Ω, the random variables e ξ k := − e α ( A k )( K − Z ) + satisfy — as e α is a solution of (5.6) — X k P (cid:16)e ξ k (cid:17) = P X k e ξ k ! = P (cid:0) − ( K − Z ) + (cid:1) . (5.7)We see that the property in (b) holds when α is as above: Y ( A ) ∼ A e Y ( A ) , ∀ A ∈ F ,as one can easily verify.(c) Let e α be the benefit sharing measure associated with e Y , so that we have for all A ∈ F : ( Z A − e Y ( A )) − = e α ( A )( K − Z ) + . Above, we have seen that whenever e Y maximal, the equality (5.7) holds, ∀ ( A k ) k finite partition of Ω. From Lemma 1, forany Q ∗ ∈ ∇P ( Z ): P (cid:0) − ( K − Z ) + (cid:1) = E Q ∗ (cid:2) − ( K − Z ) + (cid:3) = − X k E Q ∗ (cid:2)e α ( A k )( K − Z ) + (cid:3) . Using these equalities, we obtain X n (cid:0) P (cid:0) − α ( A n )( K − Z ) + (cid:1) + E Q ∗ (cid:2) α ( A n )( K − Z ) + (cid:3)(cid:1) = 0 . which proves the statement, as the sum only contains nonnegative terms. (cid:3) Theorem 3. Let Q ∗ ∈ ∇P ( Z ) and π ∗ := Z · Q ∗ ∈ e C (cid:0) c Z (cid:1) . We consider state payoffs Y ∗ = { Y ∗ ( A ) , A ∈ F } : Y ∗ ( A ) = α ∗ ( A )( K − Z ) + + ( Z ∧ K ) A , (5.8) with α ∗ ≪ P satisfying α ∗ = ( Z − K ) + E Q ∗ [( Z − K ) + ] · Q ∗ . (5.9) Then the following hold: (a) Given π ∗ , the family Y ∗ is fair. (b) Assume e Y ( A ) , A ∈ F is a family of state payoffs that is maximal and fair given π ∗ ,in the sense of fuzzy games. Then there exists b Q ∈ ∇P ( Z ∧ K ) so that P ( Z A − e Y ( A )) = P ( Z A − b Y ( A )) ∀ A ∈ F , that is, (using preference relations in Remark 6) ∀ A ∈ F e Y ( A ) ∼ A b Y ( A ) , where b Y is defined as b Y ( A ) = b α ( A )( K − Z ) + + ( Z ∧ K ) A , (5.10) with b α ( A ) := π ∗ ( A ) − E b Q [( Z ∧ K ) A ] P (( Z − K ) + ) . (5.11) roof of Theorem 3. As α ∗ is a probability measure, from Theorem 2 (a), the family Y ∗ isadmissible and maximal.Using (2.2) in (5.9) we obtain that α ∗ ( A ) := E Q ∗ [( Z − K ) + A ] P (( Z − K ) + ) . (5.12)By the commonotonicity and positive homogeneity of P and the expressions in Lemma 1: P ( Z A − Y ∗ ( A )) = P ( − ( Z A − Y ∗ ( A )) − ) + P (( Z A − Y ∗ ( A )) + )= α ∗ ( A ) P ( − ( K − Z ) + ) + P (( Z − K ) + A )= − α ∗ ( A ) E Q ∗ [( Z − K ) + ] + P (( Z − K ) + A )and P ( Z A ) = P (( Z ∧ K ) A ) + P (cid:0) ( Z − K ) + A (cid:1) . Therefore, the inequality defining fairness P ( Z A − Y ∗ ( A )) + π ∗ ( A ) ≤ P ( Z A ) , ∀ A ∈ F , writes: E Q ∗ [ Z A ] − α ∗ ( A ) E Q ∗ [( K − Z ) + ] ≤ P [( Z ∧ K ) A ] , ∀ A ∈ F (5.13)that, after replacing α ∗ ( A ) with its expression, is equivalent to: E Q ∗ [( Z ∧ K ) A ] ≤ P [( Z ∧ K ) A ] , ∀ A ∈ F . This is always satisfied, so that (a) is proved.The claim in (b) is proved as follows. First, one can check that b Y is fair given π ∗ , byreplacing in the inequality (5.13) α ∗ with b α .Let e Y be maximal and fair. From Theorem 2 (b), it is sufficient to consider the case whereits corresponding benefit sharing measure e α is a finitely additive measure. In this case, it isnot difficult to check that e Y ( A ) ∼ A b Y ( A ) if and only if e α ( A ) = b α ( A ).Therefore, it is necessary and sufficient to show that whenever e Y satisfies: (i) is fair inthe sense of fuzzy games, (ii) is maximal and (iii) its corresponding benefit sharing measure e α ( A ) is a finitely additive measure, then there is b Q ∈ ∇P ( Z ∧ K ) so that e α ( A ) = b α ( A ).Following similar steps as in the proof of (a) we find that the inequality P ( λZ − e Y ( λ )) + π ( λ ) ≤ P ( λZ ) , ∀ λ ∈ L ∞ , ≤ λ ≤ E Q ∗ [ λZ − e α ( λ )( K − Z ) + ] ≤ P [ λ ( Z ∧ K )] , ∀ λ ∈ L ∞ , ≤ λ ≤ . We define the family ( δ ( A )) with δ ( A ) := E Q ∗ [ Z A − e α ( A )( K − Z ) + ] . It can be verified that this is a finitely additive measure of total mass P ( Z ∧ K ) and theinequality (5.14) is equivalent to: Z λdδ ≤ P [ λ ( Z ∧ K )] , ∀ λ ∈ L ∞ , ≤ λ ≤ . n other words, δ is an element of the set (cid:26) ν ∈ ba (Ω , F , P ) | ν (Ω) = P ( Z ∧ K ) , Z λdν ≤ P [ λ ( Z ∧ K )] , ∀ λ ∈ L ∞ (cid:27) . We obtain that there exists b Q ∈ ∇P ( X ∧ K ), such that δ ( A ) = E b Q [( Z ∧ K ) A ]. Implicitly,we find an expression for e α : e α ( A ) = E Q ∗ [ Z A ] − E b Q [( Z ∧ K ) A ] E Q ∗ [( K − Z ) + ] , that is exactly b α . (cid:3) Fair contracts in the N + 1 players game. We consider the framework of Section4. The next theorem gives the form of some fair insurance contracts. Theorem 4. Suppose that Q ∗ ∈ ∇P ( S X ) and π i = E Q ∗ [ X i ] , for i ∈ { , .., N } . Consider the contracts { ( π i , Y i ) } Ni =1 with standard payoffs (i.e., the payoffs are as in (3.2))such that the constant proportions ( α i ) given by: α i = E Q ∗ (cid:2) X i S X ( S X − K ) + (cid:3) P [( S X − k ) + ] for i ∈ { , .., N } . (5.15) These contracts are fair and maximal. Furthermore, for the shareholders: P ( X − Y ) = 0 and k = E Q ∗ [ Y ] . Remark . We can associate with S X a probability measure α ∗ : (Ω , F ) → [0 , α ∗ := ( S X − k ) + E Q ∗ [( S X − k ) + ] d Q ∗ Then, one can verify that α i = E α ∗ " X i − Y i P Ni =0 ( X i − Y i ) ; i = 0 , , ..., N. As the probability α ∗ assigns the entire mass to the default event, we can interpret the fairbenefit share of agent i , α i as being the expected value under α ∗ of the proportion of loss indefault of agent i over the total loss in default of all agents. Remark . There is no uniqueness of the fair contracts. The fair premia in Theorem 4are known as an allocation in the fuzzy core of the game c X . Even if we consider a specificpremium vector π that is fixed, there is no uniqueness of the fair payoffs given π . In Theorem3, the general forms of fair payoffs given π was derived for the fuzzy game c Z . roof. The fact that π ∈ C ( c X ), where π i = E Q ∗ [ X i ] for all i ∈ N , is already a known resultand trivial to verify directly.By Lemma 2, Y are maximal. We now prove that Y are fair payoffs given π . As in theprevious subsection, we use the notation Z = S X + k and λ i = X i Z or X i = λ i Z, for i = 0 , ..., N. We use the notation from Subsection 5.1, in particular we take Y ∗ to be the state payofffamily defined in Theorem 3. We have Y ∗ ( λ i ) = Z λ i ( ω ′ ) Y ∗ ( dω ′ ) = E α ∗ [ λ i ]( K − Z ) + + λ i ( Z ∧ K )= E Q ∗ [( Z − K ) + λ i ] P (( Z − K ) + ) ( K − Z ) + + ( Z ∧ K ) λ i . By Theorem 3, Y ∗ = { Y ∗ ( A ) , A ∈ F } ∈ A Z, F ( K ) is a family of state payoffs that are fair inthe sense of fuzzy games, that is: P ( λZ − Y ∗ ( λ )) + π ∗ ( λ ) ≤ P ( λZ ) , ∀ λ ∈ L ∞ , ≤ λ ≤ . We recall that above π ∗ = Z · Q ∗ (see Theorem 3). We can take in the above inequality λ = λ ( S ), where λ ( S ) := P i ∈ S λ i = P i ∈ S X i Z for S ⊂ N . We obtain that P X i ∈ S ( X i − Y ∗ ( λ i )) ! + E Q ∗ "X i ∈ S X i ≤ P X i ∈ S X i ! , ∀ S ∈ N . The inequalities above write: P X i ∈ S ( X i − Y ∗ ( λ i )) ! + X i ∈ S π i ≤ c X ( S ) , ∀ S ∈ N . (5.16)We now treat separately the case with and without equity.(i) Let us assume that X = k = 0, so that we have no equity and K = k , Z = S X . Inthis particular case: Y ∗ ( λ i ) = E Q ∗ [( S X − K ) + λ i ] P (( S X − K ) + ) ( K − S X ) + + ( S X ∧ K ) λ i = α i ( K − S X ) + + ( S X ∧ K ) X i S X = (cid:2) X i + α i (cid:0) k − S X (cid:1)(cid:3) { S X ≤ k } + X i (cid:18) KS X ∧ (cid:19) { S X >k } that is exactly expression (3.2) with α i as in Theorem 4. This proves that Y i = Y ∗ ( λ i ) ∈ A X ( K ) . Then, the fact that Y = ( Y , ..., Y N ) is a fair payoff for the risk X = ( X , ..., X N )follows from the fact that π = (0 , π , ..., π N ) ∈ C ( c X ) and the above. Indeed (5.16) rites c X − Y ( S ) + X i ∈ S π i ≤ c X ( S ) , ∀ S ∈ N . (ii) Suppose k > K = k + k > k . The presence of equity introduces anasymmetry between players, as equityholders have lower priority with no paymentin case of default. In addition, the presence of equity shrinks the default event from { Z > K } to { Z > K + k } = { S X > K } . In this situation, Y ∗ ( λ i ) = Y i and even Y ∗ ( λ i ) / ∈ A X ( K ). In view of (5.16), to prove fairness of Y it is sufficient to show c X − Y ( S ) ≤ P X i ∈ S ( X i − Y ∗ ( λ i )) ! , ∀ S ∈ N . (5.17)We observe that for all Y ∈ A X ( K ) we have: {∃ i ∈ N : X i − Y i > } ⊂ ∩ i ∈N { X j − Y j ≥ } = { S X ≥ k } = { Z ≥ K } and also {∃ i ∈ N : X i − Y i < } ⊂ ∩ i ∈N { X j − Y j ≤ } = { S X ≤ k } = { Z ≤ K } . Same expressions hold true if we replace Y i with Y ∗ ( λ i ).Therefore, for any Y ∈ A X ( K ), Y vector of standard payoffs, we have X i ∈ S ( X i − Y i ) ! + = X i ∈ S ( X i − Y i ) + and X i ∈ S ( X i − Y i ) ! − = X i ∈ S ( X i − Y i ) − = X i ∈ S α i ( K − Z ) + (the last equality comes from the definition of the standard payoffs).Let us consider S ⊂ N \ { } . Using the commonotonicity of P we have c X − Y ( S ) = P X i ∈ S ( X i − Y i ) ! = P X i ∈ S ( X i − Y i ) + ! + P − X i ∈ S ( X i − Y i ) − ! = P X i ∈ S ( X i − Y i ) + ! + X i ∈ S α i P (cid:0) − ( K − Z ) + (cid:1) = P X i ∈ S ( X i − Y i ) + ! − X i ∈ S α i P (cid:0) ( Z − K ) + (cid:1) = P X i ∈ S X i S X ( S X − K ) + ! − E Q ∗ "X i ∈ S X i S X ( S X − K ) + = P ( η ( S )) − E Q ∗ [ η ( S )] . or simplicity, we have denoted above η ( S ) := P i ∈ S X i S X ( S X − K ) + . Also we haveused the expressions for α i that are given in Theorem 4. Similarily P X i ∈ S ( X i − Y ∗ ( λ i )) ! = P (cid:0) λ ( S )( Z − K ) + (cid:1) − X i ∈ S E α ∗ [ λ i ] P (cid:0) ( Z − K ) + (cid:1) = P X i ∈ S X i Z ( Z − K ) + ! − E Q ∗ "X i ∈ S X i Z ( Z − K ) + = P ( ξ ( S )) − E Q ∗ [ ξ ( S )] . We recall λ i = X i /Z . Also we denoted ξ ( S ) := P i ∈ S X i Z ( Z − K ) + .We notice that for any S ⊂ N \ { } we have that the random variable γ ( S ) := ξ ( S ) − η ( S ) is commonotonic with η ( S ). Indeed, γ ( S ) = X i ∈ S X i (cid:26)(cid:18) KS X − KS X + k (cid:19) S X >K + (cid:18) − KS X + k (cid:19) S X ∈ [ k,K ] (cid:27) and η ( S ) = X i ∈ S X i (cid:18) − KS X (cid:19) S X >K . Therefore: P X i ∈ S ( X i − Y ∗ ( λ i )) ! = P ( γ ( S )) + P ( η ( S )) − E Q ∗ [ γ ( S ) + η ( S )] == ( P ( γ ( S )) − E Q ∗ [ γ ( S )]) + c X − Y ( S ) ≥ c X − Y ( S ) . Thus we have proved (5.17) for all S ⊂ N \ { } and it remains to prove the sameholds when we include the element { } .We notice that π = k = P ( X ) and furthermore, for all S ⊂ N \ { } , the randomvariable X − Y is commonotonic with P i ∈ S ( X i − Y i ) which verifies that (5.17) holdsfor S ∪ { } whenever it holds for S . (cid:3) Conclusion on the positive role of equity and bankruptcy procedures In this paper we used cooperative game theory in order to determine the fair part of agentsin the surplus of a company. 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