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On Mutual Insurance

Science for Global Insight

International Institute for Applied Systems Analysis Schlossplatz 1 ? A-2361 Laxenburg ? Austria Telephone: ( 43 2236 ) 807 342 ? Fax: ( 43 2236 ) 71313 E-mail: publications@iiasa.ac.at ? Internet: www.iiasa.ac.at

Interim Report On Mutual Insurance


Yuri M. Ermoliev (ermoliev@iiasa.ac.at) Sjur Didrik Flam (sjur.?aam@econ.uib.no) ?

Approved by Gordon MacDonald (macdon@iiasa.ac.at) Director, IIASA January 2000

Interim Reports on work of the International Institute for Applied Systems Analysis receive only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work.

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About the Authors
Yuri M. Ermoliev is an Institute Scholar at IIASA, A-2361 Laxenburg, Austria. Sjur Didrik Fl? is the corresponding author. He is Professor at the Department am of Economics, University of Bergen, N-5007 Norway. Support from the Norge Bank and from IIASA is gratefully acknowledged.

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Owners of stochastic assets can pool their endowments to smoothen and insure individual payo?s across outcomes and time. We explore, in such a setting, how contingent shadow prices on aggregate resources can be used for three purposes: First, to design mutual contracts for risk averse agents; second, to quantify the malfunctioning of such contracts when there are risk lovers (or scale economies); and third, to estimate reasonable premiums for insurance o?ered by outside agents.

Key words: risk, insurance, mutuals, cooperative games, core, contingent prices, stochastic Lagrange multipliers, duality gap, modulus of nonconvexity, randomization.

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1 Introduction 2 Mutual insurance under concave preferences

1 2 8 9 11 11

3 Cooperation over time 4 Insuring risk lovers 5 Randomization 6 Paying for supplementary insurance


On Mutual Insurance
Yuri M. Ermoliev (ermoliev@iiasa.ac.at) Sjur Didrik Fl? (sjur.?aam@econ.uib.no) am



This paper considers several extensions of Borch’s classical study of a reinsurance market [5]. Novelties include state-dependent payo?s, multi-dimensional risks, stochastic dependence, dynamic allocations, and above all: computable core solutions. The setting is broadly as follows. Suppose individual i, when operating alone, could obtain expected payo? πi (ei) from a stochastic commodity bundle ei fully owned, delivered, produced, or handled by him. Examples are manifold. For instance, ? ei might be the randomly varying water endowments of agricultural region (or hydro-electric power station) i; ? ei could stand for nation i’s state-dependent quotas in producing diverse pollutants (or in catching various ?sh species); ? ei could account for uncertain quantities of di?erent goods that transportation ?rm i must bring from various origins to speci?ed destinations; ? ei can be the ?nancial risk, in the form of unknown monetary claims, against insurance company i. Present several such individuals i ∈ I, we consider pooling and exchange of their private holdings ei as a mean of protection against inconvenient outcomes. More precisely, we shall deal with exchange aimed at providing the concerned parties with mutual insurance. Focus will be on three problems: ? If all agents are risk averse, can they write an e?cient, socially stable, and computable contract? ? Otherwise, when some agents love risk, or if there are economies of scale, what properties will such contracts have? ? In any case, if the mutual company at hand exploits internal insurance optimally, how should it evaluate supplementary insurance o?ered by outside agents? What would be reasonable premia? To come to grips with these problems we use as benchmark the instance where all parties have concave objectives. Then, as explained in Section 2, an e?cient and socially stable contract can be fully written in terms of a contingent shadow price vector on the aggregate endowment eI := i∈I ei. That contract is, in principle, readily computable and easy to implement. It resembles a competitive equilibrium. As argued in Sections 2 and 3, it decomposes across events and time into a family of equilibria akin to spot markets.


However, and not surprisingly, when preferences are nonconcave, that sort of insurance contract, designed in terms of the said shadow prices, cannot be viable. In fact, any such contract will cause overspending and insolvency. Section 4 explains why. To mitigate these severe de?ciencies, Section 5 brie?y explores how randomization may help to restore concavity - whence enhance the prospects for good insurance. This is a companion paper to [8]. As such it elaborates on four novelties. First, we make publicity for contracts which, under concavity assumptions, are not only e?cient in the aggregate, but also robust against defection. More precisely, emphasis is here on modes of risk sharing that belong to core of a transferable-utility cooperative game. This perspective is surprisingly uncommon and far from fully explored. Second, present nonconcave preferences, we explain and quantify some basic de?ciencies of insurance contracts based on Lagrangian duality and shadow prices. Third, we indicate how randomization may facilitate the writing of good contracts. Fourth and last, we estimate the willingness of a mutual company to pay for supplementary insurance o?ered by outside agents. The paper should interest a mixed audience, comprising actuaries, economists, game theorists, operations researchers, and statisticians. All assumptions and results are stated formally and precisely. Detailed proofs are given elsewhere [8]. Here we only outline demonstrations of two key results - and rather discuss some rami?cations and seek to explain the economic signi?cance of the main propositions.


Mutual insurance under concave preferences

We accommodate henceforth a ?nite ?xed set I of risk exposed agents (or industries, regions, sectors etc.) Each individual i ∈ I owns a private, random endowment ei in some Euclidean space E . More precisely, if the state (or scenario) s ∈ S comes up, then i is fully entitled to the commodity vector ei(s) ∈ E (say, Rm if m commodities are at stake).1 One may construe these vectors as accounting for state-dependent outputs of various enterprises or natural resources. Since part of our motivation is computational, we do not hesitate in assuming S ?nite. Moreover, all agents use the same S as an exhaustive set of mutually exclusive states. In addition, the agents agree on the probability distribution p(s) > 0 of the still unknown future state s. This means that uncertainty is objective and external. It also means that the environment is una?ected by actions considered below, and there are no informational asymmetries. Note that we can allow all sorts of dependencies and associations between the stochastic vectors s → ei(s), i ∈ I. For simplicity we shall start by considering instances where uncertainty about s is fully resolved in one step. The case when information comes gradually, over several steps, will be discussed in Section 3. If individual i were to contend with his random endowment ei in splendid isolaGeneration of relevant scenarios is demanding since their number easily gets out of hand. This problem, prominent in stochastic programming, will not be explored here.


tion, he would a priori look forward to expected payo? πi (ei ) := EΠi (ei) :=

p(s)Πi (s, ei(s))


where Πi (s, ·) is his payo? function in state s. Often i can do better by collaborating with others, such collaboration implying that the holdings of the participants be pooled. Speci?cally, the members of some coalition C ? I could, in principle, join forces and compute their presumably ?nite, ex ante, aggregate, stand-alone payo? πC (eC ) := sup

πi (xi )

xi (s) =

ei (s) =: eC (s)for all s ∈ S



the aim being to distribute prospective gains among themselves.2 Since πC (eC ) is ?nite by assumption, so is the corresponding state-dependent, ex post value ΠC (s, eC (s)) := sup

Πi (s, xi(s))

xi (s) = eC (s)


for each s ∈ S. We ask: Can the grand coalition I form (be it ex ante, ex post, or both)? And if so, how might payo?s then be shared? These questions point to a family of cooperative games, all with player set I, but with various characteristic functions I ? C → vC ∈ R, and each game allowing side payments. In other words, main objects here are so-called production games, featuring transferable (maybe stochastic) payo?s. Given such a game, codi?ed by the characteristic function v = [vC ] , a payo? allocation u = (ui) ∈ RI is said to belong to its core i? u entails e?ciency: and social stability:

ui = v I , i∈C ui ≥ vC for all coalitions C ? I


Social stability means that no coalition C ? I could improve its members’ outcome by splitting away from the others. Note that mere stability is easy to achieve: Simply let the numbers (the utilities) ui be so large that i∈C ui ≥ vC , ?C ? I. Thus, not very surprising, the essential di?culty resides in the requirement that total payo? be e?cient and not distributed excessively.

Remarks: * Cooperative games do not ?gure prominently in the insurance literature, a notable exception being [2]. Borch [4], [5] also deals with cooperative games, but he never uses the core solution concept. * Creation of a mutual is, of course, only one of many ways to reduce risk exposure. Self-insurance, treaties with outside providers of insurance, and speci?c
Sup is short notation for supremum value. We could posit that all these extremal values are attained, i.e., that they be maxima. This issue will not be elaborated though because it detracts attention from the main issues.


investments to enhance safety (or reliability) are other measures. We regard these alternatives as exogenous supplements to the actions taken here. In many settings, particularly those involving fairly homogeneous communities with well spread risks, a mutual contract will be part of the overall arrangement, and worth exploring. * The insurance literature mostly considers state-independent payo?s. That optic appears reasonable for low-consequence, conventional risks such as damage on cars. It does, however, not ?t major events like severe illness or catastrophes. * Insurance theory often assumes independent or weakly associated risks. No such assumption is made here. Consequently, we cannot - and shall not - rely on any law of large numbers or central limit theorem. In fact, the subsequent analysis is applicable for major events, say catastrophes, in?icting severe and highly correlated losses. * We stress that endowments ei can be multi-dimensional. While the insurance literature, and insurance circles, often consider merely only one good, namely money, we can accommodate several ”securities”, be it ?nancial papers or real assets, these giving various sorts of dividends. Multi-dimensionality might also stem from some components of ei referring to goods available only in speci?ed combinations of state, location, and time. * Note that Πi(s, ·) is net payo? or net utility, obtainable after adverse a?ects of the ”hazard” s have been mitigated. * Admittedly, the use of expected payo?s EΠi (s, ·)is best justi?ed under repeated interaction, allowing probabilities to be estimated from observed data. Nothing precludes, however, that a mutual will be set up to protect against speci?c rare events the ”statistics” of which represents expert judgements. * Payo? Πi (s, xi(s)) = ?∞ is far from excluded. In fact, the value ?∞ represents in?nite loss (or total dissatisfaction) and accounts for violation of implicit constraints, not spelled out at this aggregate level. This abstract way of incorporating constraints is analytically very convenient (albeit not useful in computation). It helps to keep focus on some key issues. For example, we could have ? Πi(s, xi (s)) := sup Πi(s, yi ) : Li y = xi (s) with the understanding that Πi(s, xi (s)) = ?∞ whenever the equation Li y = xi (s) has no solution. Explicit representation of constraints is illustrated in Section 3. * It is tacitly assumed, when it comes to any program (2) or (3), that no concerned agent misrepresents his preferences. We posit that all functions Πi are common knowledge, or these objects can be readily synthesized, or they are reported honestly. Admittedly, this assumption is quite stringent. In the same vein, it may take some faith - or good will - to presume that all agents have the same perception of uncertainty. Coincidence of probability assessments is rare and di?cult when it comes to exceptionally important states that occur with very low frequencies. We believe, however, that iterative experiments or computations may be set up which require neither common knowledge nor equal risk perceptions, but which nonetheless converge to core solutions. * Constructions (2) and (3) are like the ones given by Shapley and Shubik [14] in their classical analysis of so-called market games.


* Note the complete absence of direct externalities in the individual objectives (1). This feature is crucial - in fact, indispensable for the subsequent analysis. It invites decomposition and decentralized decision making. To wit, if all payo? functions Πi (s, ·) are concave, then - as stated in Theorem 1 - decomposition can be supported by prices associated with contingent relaxation of the balance requirement i∈I xi (·) = eI (·). Facing appropriate prices each (presumable risk averse) agent is - as we shall see - free to make a best choice. * Equation (2) models pooling and exchange of perfectly divisible goods, freely transferable among members of C. The advantage of doing so is evident and twofold: First, aggregation o?ers increased leeway and better substitution possibilities; second, it makes possible transfers of goods across time and contingencies. Less evident is the fact that, granted concave payo?s, that is, given risk averse agents, then cooperative incentives become so strong and well distributed that the grand coalition can safely form. Its formation means that payo? can be shared in ways not blocked by any subgroup. This is stated in the following Proposition 1 (Nonempty cores ex ante and ex post) * Suppose all payo? functions ei → πi (ei), de?ned in (1), are concave. Then the ex ante payo?-sharing game [πC (eC )] becomes totally balanced. This means that the game itself and all its subgames (comprising fewer players) have nonempty cores. * Suppose state s has already happened, and that all payo? functions Πi (s, ·) are concave. Then, the ex post payo?-sharing game [ΠC (s, eC (s)] also becomes totally balanced. * If the allocations u(s) = [ui (s)] belongs to the core of the ex post game (3) for every s, then [Eui (s)] belongs to the core of the ex ante game (2).

To make these insights useful we must give some computational advice concerning how to ?nd core elements in the various settings. Denote by LC (s, x(s), λ(s)) :=

[Πi (s, xi(s)) + λ(s)(ei (s) ? xi (s))]

the ex post, state s, Lagrangian of coalition C, naturally associated to problem (3). Here and elsewhere we write simply ab for the usual inner product a · b in E . It is notationally convenient though to use the alternative inner product E(λxi ) := S s∈S p(s)λ(s)xi (s) on E . Thus, via (1), we get that LC (x, λ) := ELC (s, x(s), λ(s)) =

[πi (xi ) + Eλ(ei ? xi)]

is the standard Lagrangian associated to problem (2). Any λI ∈ E S such that supx LI (x, λI ) ≤ πI (eI ) will be named a Lagrange multiplier. Similarly, for given state s, any λI (s) ∈ E such that supx(s) LI (s, x(s), λI (s)) ≤ ΠI (s, eI (s)) will be called a contingent Lagrange multiplier. A little more notation is needed now. For any pairing ·, · (or inner product) between a Euclidean space and its dual let f? (λ) := sup {f(x) ? λ, x }



denote the conjugate of the extended real-valued function f. Note that f? is de?ned on the dual space. In terms of the customary Fenchel conjugate f ? (λ) := supx { λ, x ? f(x)} we have f? (λ) = (?f)? (?λ). De?nition (5) ?ts well to a perfectly competitive setting. Namely, if some agent buys production factors (input bundles) x at ?xed linear cost λ, x to achieve revenue f(x), then, at most, he gets pro?t f? (λ). Theorem 1. (Lagrange multipliers yield core solutions) * For any Lagrange multiplier λI the ex ante, deterministic payo? allocation ui := E(λI ei) + πi? (λI ), i ∈ I, belongs to the core of game (2). * Similarly, for any state s and associated Lagrange multiplier λI (s) the contingent, ex post, payo? allocation ui (s) := λI (s)ei(s) + Πi? (s, λI (s)), i ∈ I, belongs to the state-dependent core of game (3). * λI is an overall Lagrange multiplier i? λI (s) is a contingent multiplier for each s. Proof. Social stability obtains in the ex ante game because any coalition C receives ui = sup LC (x, λI ) ≥ inf sup LC (x, λ) ≥ sup inf LC (x, λ) = πC (eC ).
i∈C x λ x x λ

The very last inequality is often referred to as weak duality. The hypothesis concerning λI ensures strong duality: πI (eI ) ≥ sup LI (x, λI ) ≥ inf sup LI (x, λ) ≥ sup inf LI (x, λ) = πI (eI )
x λ x λ x

so that Pareto e?ciency prevails: πI (eF ) = supx LI (x, λI ) = games are in the same manner.


ui . The ex post

Remarks: * A special version of Theorem 1 was ?rst proven by Owen [11] who dealt with linear programs, plagued by no uncertainty. For extensions see [8], [13], and references therein. * Theorem 1 has a nice interpretation. Suppose contingent commodity bundles e ∈ E S were traded at a constant price vector λ ∈ E S . Then, if individual i were a price-taker, he could - at best - envisage expected pro?t πi?(λ) := supxi {πi (xi ) ? E(λxi )} . For arbitrary contingent price regime λ, given already his endowment ei, potential pro?t always dominates the fait accompli, i.e., πi? (λ) ≥ πi (ei ) ? E(λei ). Now, the particular nature of any Lagrange multiplier λI is that allocation - and pro?t considerations - can be decentralized as follows: In state s each individual i chooses a vector xi (s) such that Πi (s, xi (s)) ? λ(s)xi (s) becomes maximal. * The risk averse parties i ∈ I are under no compulsion to reach the agreement designed in terms of the multiplier. The corresponding treaty will be incentivecompatible in that no individual or group can do better alone.


* We tacitly assume that for C = I the suprema in both problems (2) and (3) are attained. Su?cient conditions for attainment are that x:

πi(xi ) ≤ r,

xi = e I


x(s) :

Πi (s, xi (s)) ≤ r,

xi(s) = eI (s)

be compact for every real r. Then the side-payments, embodied in the core allocation, can be supported by a treaty saying how the aggregate endowment eI should be split in various circumstances. Often there is no need to write that treaty though. To wit, if all objectives Πi (s, ·) are strictly concave, then two desirable things occur: First, the optimal distribution of the aggregate endowment (both ex ante and ex post) will be unique; second, the said unique choices will be made by the agents themselves. Nobody would need persuasion or coercion. In fact, the modi?ed objective Πi (s, xi(s)) ? λ(s)xi (s) of i calls forth, by itself, his seemingly agreed upon, best choice. * Theorem 1 should be seen as a generalization of Borch’s seminal study [5]. It hinges upon Lagrangian duality, exploiting weak and strong versions to generate core solutions by means of prices. Those prices are stochastic and constitute a contingent price system, as brought out in [6]. Any agent will be paid E(λI ei ) for his endowment plus the amount πi? (λI ) for his pro?t contribution (if any), both entities being computed in terms of the said price λI . The result thus resembles competitive equilibrium, giving emphasis to non-strategic, price-taking behavior and decentralized actions. * Theorem 1 begs questions whether multipliers do exist. At this juncture enters the concavity of preferences: Proposition 2 (Existence of multipliers) Suppose here that all functions Πi(s, ·) are concave and that eI (s) belongs to the interior of

{xi (s) : Πi (s, xi(s)) > ?∞} for all s. (6)

Then there exist multipliers λI and λI (s) for each s. It is informative, and helpful for economic interpretation, to relate Lagrange multipliers to marginal payo?s or shadow prices. The following result essentially derives from Danskin’s envelope theorem.3 Denote by ? the subdi?erential of convex analysis [12]. Proposition 3 (Shadow prices yield core solutions) Suppose here that all functions Πi (s, ·) are concave. Then: * λI is a Lagrange multiplier i? λI ∈ ?πI (eI ). Similarly, λI (s) is a contingent Lagrange multiplier i? λI (s) ∈ ?ΠI (s, eI (s)). * Given λI ∈ ?πI (eI ), then for any optimal x we have λI ∈ ?πi(xi ) for all i. Similarly, given λI (s) ∈ ?ΠI (s, eI (s)), then for any optimal x(s) we have λI (s) ∈ ?Πi(s, xi (s)) for all i.

It also follows from the analysis of so-called inf-convolutions in [10].



Cooperation over time

It is ?tting to elaborate brie?y on dynamic problems. For simplicity let there be only two time periods t = 0, 1, representing now and ”tomorrow”. (Extensions to more periods is, in principle, easy. It requires explicit modeling of the information ?ow though, and it comes with more cumbersome notations.) Decompose the ambient space E = E 0 ×E 1 as well as endowments ei (s) = (ei0 , ei1)(s) into corresponding stage-relevant parts. Most important, since s is unknown at time 0, we require that the time 0 component eio (s) be constant as a function of s. This restriction is commonly referred to as non-anticipativity: Future knowledge cannot be exploited before it comes about. We now de?ne i′ s problem as follows. If he uses an information-adapted strategy s ?→ yi (s) = (yio , yi1(s)), he enjoys state-dependent, time-separable payo? fi (s, yi ) := fi0 (yio ) + fi1 (s, yi1(s)) provided the constraints gi (s, yi) := [gio (yio), gi1 (yio , yi1 (s))] ≤ ei (s) := [eio, ei1 (s)] for all s ∈ S. are satis?ed. Otherwise his payo? is ?∞. (As usual, inequality between vectors is meant to hold coordinatewise.) De?ne πi(ei ) to be the maximum expected value of this program. One may easily argue that πC (eC ), as de?ned in (2), assumes the alternative form πC (eC ) = sup
y i∈C i∈C

Efi (s, yi) :

gi (s, yi ) ≤ eC (s) for all s .

Let here LC (y, λ) := E

[fi (s, yi ) + λ(s) {ei (s) ? gi (s, yi )}] .

Proposition 4 Suppose πI (eI ) ≥ supy LI (y, λI ) for some Lagrange multiplier rule s → λI (s) = (λI0 , λI1 (s)) ≥ 0. Then, paying each i the amount ui := sup E [fi (s, yi ) + λI (s) {ei (s) ? gi (s, yi )}] ,

yields a core allocation ex ante. Ex post, at time t = 1, in state s, with already sunk optimal decisions yo , the remaining game with conditional payo?s ΠC (s, eC1(s)) := sup
y1 i∈C

fi1 (s, yi1) :

gi1 (s, yi ) ≤ eC1 (s)

admits an ex post core allocation ui(s, yi0 ) = sup [fi1(s, yi1 ) + λI1 (s) {ei1(s) ? gi1 (s, yi )}] .

As above, concavity - and some constraint quali?cation - su?ces to guarantee existence of multipliers. In particular, it would be enough for (6) to have all functions fi (s, ·) and (components of) ?gi (s, ·) concave ?nite-valued, and that each strict inequality gi (s, ·) < ei (s) be solvable. This is the so-called Slater condition. Numerical techniques for solving such stochastic programs are presented in [7].



Insuring risk lovers

Insurance has positive value to risk averters. They are willing to pay for greater security. Mathematically, this phenomenon manifests itself here as an equality between two extremal quantities. These are, on one side the presumably ?nite, e?cient payo? P := πI (eI ), and on the other side, the optimal value D := inf λ supx LI (x, λ) of the associated dual program. Theory tells that under (6) the equality P = D, so conducive for computation and decomposition, does indeed obtain provided the (presumably continuous) aggregate objective i∈I πi(xi ) is concave. In other words, aggregate risk aversion su?ces for the design of the contracts described above. Absent aggregate risk aversion (i.e., absent aggregate concavity) that design no longer works well. The Lagrangian LI (x, λ) could then have no saddle value: P := πI (eI ) = sup inf LI (x, λ) < inf sup LI (x, λ) = sup
x λ λ x λ i∈I

[Eλei + πi? (λ)] =: D

and there would be a positive so-called duality gap d := D ? P. That gap determines how well solutions to (4) can be approximated:

Theorem 2. (Approximate core allocations) Suppose (6) holds. Then there exists some λI ∈ E S which maximizes i∈I [Eλei + πi? (λ)] . Any such λI de?nes an allocation ui := EλI ei + πi? (λI ), i ∈ I, which is socially stable in so far as ui ≥ πC (eC ) for all coalitions C ? I.

Moreover, the grand coalition will overspend with excess d: ui = πI (eI ) + d.

Proof. The ”primal value” P = πI (eI ) is ?nite by assumption. Let λI be any optimal solution to the dual problem inf λ i∈I [λei + πi? (λ)] . Such a solution is known to exist under (6). Then evidently, i∈I ui = supλ i∈I [λei + πi?(λ)] = πI (eI ) + d, and ui =
i∈C i∈C

[λI ei + πi?(λI )] = sup LC (x, λI ) ≥

inf sup LC (x, λ) ≥ sup inf LC (x, λ) = πC (eC )
λ x x λ

for all coalitions C ? I. Theorem 2 says that if some outside benefactor would contribute d on the condition that coalition I forms, then cooperation could indeed come about. If that transfer does not come, the mutual company faces bankruptcy. It cannot, in the average, honor the contracts. Some bills will be left unpaid, and contingent plans may be hard to implement.

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This result may - at ?rst glance - inspire some sadness or frustration. In mathematical terms is shows the limitation of convex analysis. In economic terms it stresses that insurance becomes quite di?cult under economies of scale. That feature is certainly not surprising. It merely expresses that risk lovers require compensation for getting rid of uncertainty. On second thoughts Theorem 2 provides some useful insights. We speculate brie?y about these next. First, one may argue that when d > 0, the core is likely to be empty. Risk loving members of I will shy away from insurance. They ?t the company badly and should, if possible, be excluded. Alternatively, Theorem 1 can be seen as making a case for self-insurance of some parties: sectors or individuals enjoying increasing returns should live without some sorts of insurance. They probably will not join the mutual; they are foreign parties to the community I. In particular, this could apply to public utilities and to holders of options which are subject to price uncertainty. Theorem 2 could also be stretched as an argument for collective rescue operations: some rare ”disasters” call upon society at large to mitigate the post-event consequences. It might also happen that insurance, as designed here, improves so much on e?ciency, that a tax d can justi?ably be levied on the members of I (and on others maybe). Anyway, to quantify the disheartening de?ciency d, that number must be related to data. To simplify this task we suppose that all domains of functions are convex.4 Then, following Aubin and Ekeland [1], given any function f from a real vector space into R∪{±∞} with convex e?ective domain domf := f ?1 (R), we measure that function’s lack of convexity by the number ρ(f) := sup f(

αk xk ) ?

αk f(xk ) ,

the supremum being taken over all ?nite families αk ∈ [0, 1] , xk ∈ domf, k∈K αk = 1. Clearly, ρ(f) ≥ 0, ρ(f+g) ≤ ρ(f)+ρ(g), ρ(f) = 0 ?? f is convex, and the largest convex function convf ≤ f must satisfy f ? ρ(f) ≤ convf. Suppose henceforth that all payo? functions πi (·) are upper semicontinuous (usc for short). The following result derives from [1]: Proposition 5 (Estimating the duality gap) With πI (eI ) ?nite suppose (6) holds and that domπi := {xi : πi (xi ) > ?∞} is nonempty convex for every i ∈ I. Then d ≤ i∈I ρ(?πi ). Let henceforth E = Rm. Using the Shapley-Folkman Lemma Aubin and Ekeland [1] proved Proposition 6 (A tighter estimate on the duality gap) With πI (eI ) ?nite and (6) in vigor, suppose E = Rm and that domπi is convex for all i ∈ I. Then d ≤ (m + 1) maxi ρ(?πi ).

For discrete domains see [3].

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Remark: The core incorporates much stability and facilitates the study of many games with side-payments. It can be either empty though or so large as to loose predictive power. But insurance (production) games are somewhat di?erent and at advantage: Granted concave preferences the core contains price-supported elements. And absent concavity, but present many players, the core comprises computable and good approximations. For large games it is widely known that cooperation and competition approximate each other well; they can nearly be reconciled [9], [15], [16], [17]. This can also be brought out here by letting I be a nonatomic measure space and invoking the Lyapunov convexity theorem.



To mitigate the de?ciency d under nonconcave preferences one might proceed as follows. Suppose agent i is restricted to apply a ?nite set Fi ? E S of strategies. Presumably ei ∈ Fi . Let Xi be the set of probability distributions (the simplex) over Fi and de?ne (with slight abuse of notation) Πi (s, xi(s)) :=
fi ∈Fi

xi (fi )Πi(s, fi (s))

and, as before, πi (xi) :=

p(s)Πi (s, xi(s))

when xi ∈ Xi , ?∞ elsewhere. Evidently, concavity (in fact, linearity) of objectives now obtains. A constraint of the sort i∈C xi (s) = eC (s) now means that xi (fi )fi (s) = eC (s).
i∈C fi ∈Fi

So, one would use the convention xi (s) := fi ∈Fi xi(fi )fi (s) and, modulo this rule, the analysis of Section 2 applies. Admittedly, implementation of randomized devices is not straightforward, and to ?nd a practical form appears much of a challenge. In some cases implementation may amount to maintenance of activities that yield inferior return, but o?er good recourse options when such are needed. In other cases, randomization can take the form of part-time utilization of alternative production lines, technologies, strategies, or modes of behavior.


Paying for supplementary insurance

Suppose some outside agent o?ers to add the some random vector e to the existing ? aggregate eI . This means that in state s he would transfer the resource bundle e(s) ? to the mutual company. How would that company price such an o?er? Clearly, a reasonable price would equal the total value added, namely πI (eI + e) ? πI (eI ). Let ?

– 12 –

? λI denote a Lagrange multiplier, if any, under the new aggregate endowment eI + e. ? An ex ante core allocation would then be to pay i the amount ? ? ui = E(λI ei ) + πi?(λI ) ? ? and let the outside insurer have E λI e. If e is only a minor addition to the aggregate ? endowment, then, under quali?cation (6), πI (·) is likely to be Lipschitz continuous near eI . Any dual optimal solution λI will then be a generalized gradient of πI (·), and E(λI eI ) becomes a reasonable estimate of πI (eI + e) ? πI (eI ). ?

[1] J. P. Aubin and I. Ekeland, Estimates of the duality gap in nonconvex optimization,
Mathematics of Operations Research 1, 225-245 (1976).

[2] B. Baton and J. Lemaire, The core of a reinsurance market, The ASTIN Bulletin. [3] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York (1982).

[4] K. H. Borch, Reciprocal insurance treaties seen as a two-person cooperative game,
Skandinavisk Aktuarietidsskrift 29-58 (1960).

[5] K. H. Borch, Equilibrium in a reinsurance market, Econometrica 424-444 (1962). [6] K. H. Borch, The Economics of Uncertainty, Princeton University Press, New Jersey

[7] Y. M. Ermoliev and R. J-B. Wets, Numerical Techniques for Stochastic Optimization,
Springer-Verlag, Berlin (1988).

[8] I. V. Evstigneev and S. D. Fl? Mutual insurance and core solutions, manuscript. am, [9] M. Kaneko and M. H. Wooders, The core of a game with a continuum of players
and ?nite coalitions: The model and some results, Mathematical Social Sciences 12, 105-137 (1986).

[10] P.-J. Laurent, Approximation et optimisation, Hermann, Paris (1972). [11] G. Owen, On the core of linear production games, Mathematical Programming 9,
358-370 (1975).

[12] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton (1970). [13] M. Sandsmark, Production games under uncertainty, to appear in Computational
Economics (1999).

[14] L. S. Shapley and M. Shubik, On market games, J. Economic Theory 1, 9-25 (1969). [15] D. Sondermann, Economies of scale and equilibria in coalitional production
economies, Journal of Economic Theory 8, 259-291 (1974).

– 13 –

[16] M. H. Wooders and W. R. Zame, Large games: Fair and stable outcomes, J. Economic
Theory 42,59-93 (1987).

[17] M. H. Wooders, Equivalence of games and markets, Econometrica 62, 1141-1160


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