Robust insurance mechanisms and the shadow prices of information constraints

. We consider a risky economic project that may yield either profits or losses, depending on random events. We study an insurance mechanism under which the plan of project implementation maximizing the expected value of profits becomes optimal almost surely. The mechanism is linear in the decision variables, "actuarially fair" and robust to changes in the utility function. The premium and the compensation in the insurance scheme are expressed through dual variables associated with information constraints in the problem of maximization of expected profits. These dual variables are interpreted as the shadow prices of information. Along with the general model, several specialized models are considered in which the insurance mechanism and the shadow prices are examined in detail.


Introduction
Problem statement.We consider the following problem of decision-making under uncertainty.A decision-maker (manager) has to carry out a risky project during a time interval [to, t l].At time to he chooses a plan (decision) x E X which specifies how the project will be realized.The consequences of the decision x depend on random events.At time to the future course of these events cannot be predicted with certainty; however, the probabilities of the possible outcomes are known.By the end of the time period [to,t1], full information about the stochastic factors which might influence the realization of the project is available.Random outcomes may be both favorable and unfavorable.In case of favorable outcome, the project yields profit, while an unfavorable course of events leads to losses.Suppose that the manager works for an organization or firm which performs a large number of similar projects.If the organization as a whole is risk-neutral, it I.V. EVSTIGNtEV, W.K. KLEIN HANEVELD AND L.J. MII=tMAN would want each of its managers to maximize expected profits.However "if the reward of the manager depends in some measure on his observed profits and if he is a risk-averter, he will wish to play safe by following a course which leads to more predictable profits, even if the expected value is lower.To avoid this outcome, the organization should provide insurance against unfavorable external contingencies..." (Arrow [3]).
Insurance.In the present paper, we describe an insurance mechanism that mo- tivates the manager to make a decision, E X, yielding the maximum expected profit.We consider an insurance scheme under which the decision appears to be optimal for the manager at almost all random situations.Suppose that before making a decision the manager computes possible values of his income, assuming one or another particular realization of future random events.Suppose he takes into account the insurance premium he would pay as well as the compensation provided by the insurance mechanism.Having performed these computations for various admissible decisions z E X and various states of the world, he will conclude that for almost all random outcomes the decision yields the maximum reward.Thus, the manager will practically never regret that he has chosen the plan , rather than some other feasible plan.
We assume that X is a convex set and the manager's payoff function is concave in x.Under these conditions and certain technical assumptions, we prove the existence of a linear insurance mechanism possessing the above described property (in this mechanism the premium and the compensation are linear functions of x).Having established the general existence theorem, we then apply it to some specialized models.In those models the insurance scheme we deal with is examined in more detail.In particular, we investigate the random variables describing the payoffs with insurance and without it.We find conditions under which the insurance mechanism "stabilizes" the random payoff in the sense of one or another criterion.Robustness and linearity.Various aspects of the general economic problem considered in this paper have been analyzed by many authors.One can point, for example, to studies of optimal insurance (Arrow [3], Drze [8], Sorch [7]) and of the principal-agent problem (Grossman and Hart [11], Hart and nolmstrbm  [13]).
For the most part, the models examined in the literature are described in terms of the individual's utility functions of money.We formulate the problem and give a solution to it without using these functions.The study is addressed to economic situations in which there is no reliable information about individual utilities.If such information is available, then one can employ the conventional methods leading to the construction of an optimal insurance system.It can be easily shown, however, that if we wish to deal with the class of all (state-dependent, increasing) utility functions, then the only way to implement the decision $ by means of a linear insurance scheme is to use the mechanism described above.We call this mechanism robust, since it is insensitive to utility changes, and since the desired implementation property is retained over the whole class of utility functions.
We concentrate on the class of linear insurance schemes because such schemes are often used in practice, because they are often cheaper to implement and to compute, z (2) Here Ep(s)is defined as (Epl(s),...,Epn(s)).Condition (2) means that the premium is actuarially fair.Assumption (2) is, of course, an idealization.One can consider more realistic insurance schemes satisfying the condition = E p(s) 4-/9, where is a (relatively small) positive number.Insurance mechanisms of this type will be examined in our next paper.Here, we restrict attention to the case 9 0. This enables us to simplify the form of presentation and to concentrate on the most essential features of the class of mechanisms under consideration.
The central result.We will analyze the insurance schemes described in Theorem 2.1 below.THEOREM 2.1 There exists an insurance scheme (/5,p(.))such that = Ep(s) and for l-almost all s E S, the inequality f(s, x) x 4-p(s)x <_ f(s, ,) fg, 4-p(s) (3) holds for all z , X. Relation (3) states that under the insurance mechanism (/5,p(.)) the decision is optimal in almost all random situations.Recall that stands for the decision maximizing E f(s, z) over X.It should be noted, however, that ifis any decision and (/5, p(-)) any insurance scheme satisfying (2) and (3), then is necessarily a maximum point of E f(s,z).To prove this take the expectations of both sides of (3) and use (2).By employing (2), we also conclude that the expected reward with insurance is the same as without it: E f(s, z) = E(f(s, ) z + p(s)z) for all z X.Some basic properties of (,p(.)).Let (,p(.))be an insurance scheme pos- sessing properties 2) and (3).Fix any vector b R n and define { + b, q(s) = p(s)+ b.Then the insurance scheme ({,q(.))will satisfy (2) and (3) as well.This means, first of all, that the insurance mechanism under consideration is not unique.Furthermore, the vectors ff and p(s) are not necessarily positive.
The uniqueness and the positivity of the vectors and p(s) can be established only if these vectors are appropriately normalized.For example, suppose that s takes two values: s = 0 (failure) and s = 1 (success).Then a ntural normalization is given by the condition p(1) = 0, which means that in ce of success there is no compensation.In specialized models that will be considered in Section 5 this condition together with some additional assumptions guarantees the nonnegativity and the uniqueness of and p(s).In Section 4, n example will be presented in which and p(s) are nonnegative but not necessarily unique, even under the above normMization.
It should be noted that negative values of and pi(s) have an edonomic meaning well.For example, components of the vectors ff and p(s) may be negative if insurance is combined with a lon (this is one of the oldest forms of insurance used in the marine practice, see Borch [7]).Suppose int X and the function f(s, z) is differentiable at the point .Then inequality (3) yields 9O I.V. EVSTIGNEEV, W.K. KLEIN HANEVELD AND L.J. MIRMAN p(s) f' (s, ) almost surely (a.s.) (4) where f' (s, z) is the gradient of f(s, z) with respect to z.Thus, the function -p(s) is essentially unique: one can change it only on subsets of S having measure zero.
Consider for the moment the one-dimensional case n 1 and X C R1.In this case, inequality (4) means that the difference between the premium price and the compensation price is equal a.s. to the marginal income f'(s, ,) for the decision .
In certain models, we may define favorable (resp.unfavorable) random outcomes as those for which the marginal income is positive (resp.negative).Suppose (4)   holds for each s 6 S.Then, we can conclude that the compensation exceeds the premium if and only if the random outcome is unfavorable.
Utility and optimal insurance.Let us discuss the relationships between our approach and the conventional theory of optimal insurance using utility functions (e.g.[7]).Denote by//the class of all real-valued functions V (s, r) of s 6 S and r (-oo, +oo) satisfying the following conditions: V(s, r) is measurable in s; U(s, r) is continuous and non-decreasing in r; for any function ( S --+ (-oo, +e) with E[(s)[ < oo, the expectation E[U(s,((s))[ is finite.Let (/5,p(.))be an insurance scheme possessing property (3).Then we have maxE U(s, f(s z) z + p(s)z) E U(s f(s, 2) y: + p(s).)zEX for any function U 6/d. Suppose that a function U(s,-), belonging to the class/d, represents the decision- maker's (manager's) utility of money at state s.If the goal of the manager is to maximize the expected utility, he can achieve this goal by taking the decision , as relation (5) shows.In this sense, the insurance scheme (/5, p(.)) makes it possible to implement the decision for any utility function in the class//; therefore we call this scheme robust.It is important to note that the choice of is optimal regardless of the concavity or convexity of U(s, .)(i.e., regardless of the manager's attitude toward risk).
Conversely, suppose that (5) holds for all U /,/.Then we can apply (5) to any function U of the form U(s,r) = u(s).r,where u(s) is measurable, positive and bounded.We have E u(s)[f(s, z) z + p(s)z] < E u(s)[f(s, ,) + p(s)] for each z 6 X.This implies + _< + e x.
Since f(s, z)-z + p(s)z is continuous in z, we conclude that with probability one inequality (3) holds for all z X.Thus, the truth of ( 5) for all U 6/d is equivalent to the truth of (3).
If we fix some particular utility function U(s, r), then we can construct an in- surance scheme which implements the decision and yields, in general, a greater expected utility than the insurance scheme (/5,p(.)).Consider a simple example.
Let U(r) be a utility function independent of s.Suppose U(r) is strictly concave, increasing and continuous.Assume that X is an interval in [0, ) and is a strictly positive number.Let (, q(.)) be an insurance scheme such that q(s) (E f(s, ) f(s, ))/.
Mathematical results related to Theorem 2.1.Theorem 2.1 can be proved by various ways.For example, one can derive it from the following general fact in convex analysis.
Recall that a linear functional b on R is called a support functional of a concave function (x),x X (C_ R), at the point x0 X if (x)-(xo) < b(x-xo), x X.The set of all such functionals is denoted by 00(-(= O(xo)).THEOttEM 2.2 Let f(s, x) be a function defined for s in a measurable space S and for x in a convex compact set X C_ Rn.Let # be a probability measure on S. Suppose f(s, x) satisfies conditions (i)-(iii).Then we have f f s s Theorem 2.2 follows from a statement in Ioffe and Tihomirov [15], Section 8.3, Theorem 4. (The statement cited deals with the situation when the domain of f(s, x) depends on s and therefore the result has a more complex form.)A version of the above formula involving conditional expectations is proved in Rockafellar and Wets [22].The earliest reference to results of this type is, apparently, Ioffe and Tihomirov [14].
To obtain Theorem 2.1 as a consequence of Theorem 2.2, observe the following.
Since is a point of maximum of El(s, x), we have 0 toe f f(s, .)(ds).By virtue of Theorem 2.2, there is an integrable vector function l(s) such that l(s) cOef(s, .)a.s. and 0 f l(s)p(ds).Consider any i5 and p(s) satisfying 15-p(s) =/(s).Then the relations = Ep(s) and (3) hold, which proves Theorem 2.1.
In the next section we will deduce Theorem 2.1 from other results (dealing, gener- ally, with not necessarily integral functionals).This will yield an independent proof of the theorem and make the paper self-contained.Furthermore, the course of argu- mentation will point out important relations between robust insurance mechanisms and the shadow prices of information.
3. Insurance prices and Lagrange multipliers for information constraints Information constraints.In this section we regard the problem (P) as a stochas- tic optimization problem with an information constraint.This constraint is rep- resented as a linear equation in the space of decision functions.We define p(.) as a Lagrange multiplier (in a function space) associated with this constraint.We show that (Ep(.), p(-)) is an insurance scheme satisfying the conditions described in Theorem 2.1.
Denote by X the class of all measurable mappings x S ---> R n such that x(s) E X a.s.We call these mappings decision functions (or strategies).Define F((.))--E f(s, (s)), c(.)E X.The problem (7) can be written as (P) Maximize F(z) over all vectors z E X.
Together with this problem we consider the following one: (Pl) Maximize F(x(.)) over all functions x(-) E X.
In he latter problem, the functional F(.) is maximized over all possible strategies x(s).This means that the decision z(s) is made with full information about s.It is clear that in order to solve (Pl), we have to maximize f(s, z) over X for every fixed s (taking care of the measurability of the solution obtained).In contrast to (:Pl), the problem (P) deals with the maximization of the functional F(-) over all constant (or a.s.constant) functions z(.) X', which can be identified with vectors z X.In this case, the decision z is taken without information about s.
The problem (P) can be obtained from the problem (:Pl) by adding the constraint: z(s) is constant (a.s.).
This constraint can be represented in various equivalent forms.Let us write it in the following form = E (8) Clearly ( 7)implies (8) and vice versa.Thus, the problem (P)is equivalent to the following problem: Maximize F(m(.)) over all functions z(.) X satisfying the constraint ()-E (-) = 0 (.s.) ROBUST INSURANCE MECHANISMS

93
The sets of solutions of (79) and (790) coincide.Consequently, the decision , which we defined in the previous section, is a solution to both of these problems.
The equivalent requirements (7)-( 9) express the fact that the decision x(.) is made without information about s, and so these requirements may be called information constraints.Equation (9) has the form B[z(-)] -0 (a.s.), where B is the linear operator transforming a function z(s) into the function z(s) E z(-).Thus, (9) is a linear operator constraint.Theorem 3.1 below shows that there exists a Lagrange multiplier p(.) removing this constraint.
As in Section 2, conditions (i)-(iv) are assumed to hold.THEOREM 3.1 There exists a measurable function p S -+ R such that Elp(s)l < and + E < (11)   for all z(.) 2,.
Let us deduce Theorem 2.1 from Theorem 3.1.
REMARK 3.2 If S consists of a finite number of points, then Theorem 3.1 can easily be obtained by usiiag a standard version of the Kuhn-Tucker theorem for convex optimization problems with linear equMity constraints (see, e.g., Ioffe and Tihomirov [15], Section 1.3.2).Under the assumption that S is a Borel subset of Rm, Theorems 3.1 and 2.1 follow from results of Kockafellar nd Wets [21], Section 4. Rockafellar and Wets developed a theory of Lagrange multipliers for information constraints, dealing with multistage stochastic optimization problems.
A more general result.Theorem 3.1 is a special case of the following, more general, result.
THEOI%EM 3.2 Let F(x(.)), x(.) E X, be any concave functional continuous with re- spect to a.s.convergence.Then for this functional the problem (TPo) has a solution, 2, and there exists a measurable function p(s), satisfying (10) and (11).
Recall that X {z(.) z(s) E X a.s.},where X C_ R is a compact convex set with int X .I n the above theorem, it is not assumed that the functional F(.) is representable in the form (6). If F(z(.))E f(s,z(s)), where f(.,-) possesses properties (i)-(iii), then F(-) is concave and continuous with respect to a.s.con- vergence.Thus, Theorem 3.1 follows from Theorem 3.2.FunctionMs which are not necessarily representable in the form (6) will be considered in Section 7.
A proof of Theorem 3.2 is contained in a paper by Evstigneev [10], Theorem 1, where an analogous result is established for more general (multistage, discrete time) stochastic optimization problems.See also Back and Pliska [4], where con- tinuous time analogues of the above theorem and their economic applications are discussed.For the reader's convenience, we present a direct proof of Theorem 3.2 in the Appendix. 4. A linear model with two states Model description.Let us investigate the insurance mechanism under consid- eration in the simplest possible case: S consists of two points, 0 (failure) and 1 (success); X is a segment [0, a] in the real line; the function f(s, z) is linear in x, i.e., :(0, = = where q0 and qt are fixed numbers.We assume that q0, ql and a are strictly positive.In this model, the set X of decisions is a set of real numbers, the segment [0, hi.A number z X may be interpreted as an "intensity" of realization of the risky project.In accordance with the interpretations mentioned in Section 2, z may represent the amount of money invested in an enterprise, or the amount of commodity shipped under a risky transportation, or the area of cultivated land in a model of agricultural insurance, etc.In case of success, the project yields profit qtz; failure leads to losses qoz.The probabilities of the random outcomes s = 0 and s = 1 are zr0 > 0 and rl > 0, respectively.
We have This function is linear in z, and so there may be three possible sets of maximum points of F(z) on [0, a]" {0}, [0, el, and {a}.Of interest for us is the last case, when F(z) has a unique maximum attained at .= a.This is so if and only if r0q0 < rt qt. ( ROBUST INSURANCE MECHANISMS 95 Under condition (14), the maximum expected payoff corresponds to the maximum intensity level of the project: = a.Throughout this section, inequality (14) will be supposed to hold.Characterizing the robust insurance schemes.Let us describe all the insur- ance schemes (6,p(.))satisfying requirements (2), (3) and the following additional condition v() =0.() According to this condition, if the random outcome is favorable, then the compen- sation equals zero.We have z = zp() 0p(0)+ .0=0;(0), which means that the premium price is equal to the compensation price times the probability of failure.Writing inequality (3) for s = 0 and s = 1, we find z E [0, hi.Recall that = r0p(0).In view of this, inequalities (17) and (18) are equivalent to the following ones -q0 0(0) + (0) > 0, q-r0p(0) > 0.
Therefore at least one such insurance scheme exists (which also follows of course from the general result, Theorem 2.1).
Inequalities (19) and (20) have a clear economic interpretation.By virtue of (19), p(O)z < qoz + z, z X.Thus, in case of failure the compensation p(O)z refunds both the amount of losses, qoz, and the premium payed, ihz.In view of (20), we have z < qz, z X, and so the value of the premium does not exceed the value of the profit yielded by the project in the case of success.Payoff with and without insurance.Consider the random variables which specify the random income with and without insurance, respectively (computed for the decision $).Recall that " = a, nd hence r/(s) = -qoa if s = 0 and y(s) = qla if s = 1.Further, we have ,(0) = --qla rop(O)a + p(O)a = --qoa + wlp(0)a = (rlp(0) qo)a ( 25) and (1) qla-rop(O)a -(ql rop(O))a.
(26) In view of ( 21), (0) _> 0 and (1) _> 0. Thus, each of the insurance schemes satisfy- ing conditions (22) guarantees that the amount of money left after the realization of the project is nonnegative (the possibility of bankruptcy is excluded).Three special cases.Let us examine the following three cases" 1) p(0) qo/rl; 2) p(0) q0 + ql; 3) p(0) qz/ro.In the first case, the compensation p(O)a and the premium i5,-r0p(0) take on minimal admissible values (see (21)).In the last case, the premium and the compensation are maximal.The insurance scheme with p(0) q + q0 has a special property which will be discussed later.Observe that the value p(0) q +q0 is admissible, since each of the inequalities qo/r < qo +ql, qo + qz < ql/wO is equivalent to (14).Case 1 ("normal").If p(0) qo/rl, then (s) 0 for s 0 and (ql qoro/rl)a for s 1 (see (25) and ( 26)).It follows from ( 14) that 5(1)is strictly positive.Comparing the random variables (s) and y(s), we see that y(0) -qoa < 0 (0) < (1) < (1).Thus, insurance increes the smallest value of income and reduces its greatest value.The random variable (s) is "less variable" than y(s) in any reasonable sense.In particular, the variance of (s), Vat E 2-(E) 2 (rO/l)a2(rlql 0q0)2, (27) is strictly less than the variance of (s), which is equal to ola2(ql + qo)2.Case 2 ("ideal").If p(0)-q0+ql, then (0)-( 1)-(iqi-oqo)a E f(s, ).This means that the value of (s) in the ces of success and failure coincide.Such an insurance system leads to a. complete stabilization of income.To achieve this, one h to pay the premium fi-0(q0 +ql)a greater than the premium fi-o(qo/l)a considered in ce 1. Case 3 ("exotic").If p(0) ql/o, then the premium and the compensation achieve their maximal possible values.In this exotic case, we hve # = qi, and so all the profit from the successful realization of the project is spent for the premium.
By computing the random variable (s), we obtain (s) ((l/ro)ql-qo)a if s 0 and (s) 0 if s = 1.It follows that the random income (s) is equal to zero in ce of success and is strictly positive in case of failure.Thus, insurance turns the former unfavorable random outcome into a "pleant surprise".The variance of (s) equals Var orla2(qll/ZOqo) 2 (l/ro)a2(lqloqo)2, (28) and we can see from (28) that Vat > rOrla2(ql+qo) 2 Var r ifr0 < ql/2(qo+q).
Therefore the insurance mechanism with p(0) = q/ro increases the variance of income if the probability r0 is small enough.This kind of "pathological" phenomena related to insurance is well-known.For example, widely used systems of automobile insurance cannot exclude cases when an indemnity for a stolen old car may exceed its real market value.Further discussion of this topic would lead us to the problems of moral hazard and adverse selection, that are, in general, beyond the scope of the paper (e.g.see

Milgrom and Roberts
[S]). 5. Nonlinear models with two states Assumptions.In this section, as in the previous one, we assume that the state space S consists of two points, 1 and 0 (success and failure).The random parameter s takes the values 0 and 1 with probabilities r0 > 0 and r > 0. The set X coincides with the segment [0, a] (a > 0) in the real line.However, in this section the function f(s,.) is not supposed to be linear.
We assume that for each s E {0, 1}, the function f(s, ) is continuous and concave in z E [0, el.Then f(s, z) satisfies conditions (i)-(iii) (see Section 2).Clearly, the set X = [0, a] satisfies (iv).Let be a point in [0, a] at which the function attains its maximum.We impose the following assumptions on f(-, .)and : (f.0)For each s S, we have f(s, O)-O.
(f.1)The point belongs to the interior (0, a) of the segment [0, a], and the function f(s,-) is differentiable at z for any s q S. (f.2) We have f' (1, ) According to (f.0), the decision 0 (inaction) leads to zero payoff at every state s S. Assumption (f.2) means that the marginal profit yielded by the plan in case of success is strictly positive.By using (f.0), (f.2) and the concavity of f(1,-), we conclude f(1,) > 0. Hence, in case of success the realization of the project at the intensity level gives a strictly positive profit.Writing a necessary condition for an extremum of the function (29) at the point ', we find -0f'(0, + f' (1, 0.

(3O)
Consequently, f'(0, 5) < 0, (31) and so the marginal reward in case of failure is strictly negative.Also, we have E f(s, ) > O, since the function E f(s, z) is concave and satisfies E f(s, 0) 0 and dE f(s,) 0 Thus, the expected profit corresponding to the optimal decision dx is positive or equal to zero.
i.v.EVSTIGNEEV, W.K. KLEIN HANEVELD AND L.J. MIll.MAN We do not assume that the payoff f(0, ') in case of failure is necessarily negative.
Formally, this assumption is not needed for the validity of the results we obtain.
However, when we say that in case of failure the implementation of the plan " leads to losses, we have in mind the number f(0,') is negative.The absolute value of this number, If(0, )1= -f(0, '), specifies the size of losses.

(39)
The first of these two inequalities shows that the premium/, is not greater than the profit f(1, ) in case of success.By virtue of the second inequality, the compensation p(0)" covers both the premium paid and the amount of possible losses, so that the resulting net payoff is nonnegative.Relations (38) and (39) are similar to those we have established in the linear case (see (19) and ( 20)).Note that the only assumption we used when proving (38) and (39) was (f.0).
In the linear model with two states, we have established the existence of a whole family of robust insurance schemes differing from each other in their properties.In the nonlinear model under consideration, the robust insurance mechanism defined by (32)-(34) is unique.However, the properties of this mechanism may be quite different for different functions f(-, .)emd probabilities r0 and r.We shall show this in the course of our further study in the remainder of the present section.
We will examine conditions under which one or another of the following relations holds: (0) < ( 1 The first of these relations expresses a natural requirement on the insurance mechanism: the payoff (0) in case of failure should not be greater than the payoff (1)  in case of success.If (46) holds, then (s) does not depend on s, and so insurance leads to the complete stabilization of the payoff.Inequality (47) is equivalent to the following one: Var < Varr/, since Var = 7r07r[(0)- (1)[ and Varr/ = 7r0r[ r/(0)-r/(1)[.Thus, (47) is satisfied if and only if the insurance mechanism reduces the variance of the payoff (for the decision ).
If (45)is true, then by using (44), we find /(0) < (0) < (i) < r/(1), which yields (47) with strict inequality: Observe that properties (48) and E E y imply E u() > E u(r/) for all concave u: R -+ Rz. (5O) Proof of this implication, which is not hard to obtain, is left to the reader.A more general result will be established in the next section (see Theorem 6.1).Prop- erty (50) means that every risk-averter would prefer the random payoff to the random payoff /.In this sense, is "less risky" than r/ (e.g.see Rothschild and Stiglitz [23]).
Conditions for risk reduction.In general, none of relations (45)-(47) follows from hypotheses (f.0)-(f.2),which we suppose to hold in the present section.To guarantee the truth of these relations, one has to impose additional conditions on the model.Such conditions are described in Theorem 5.1 below.
(58) The functions -g(x) and h(x) g(x) are supposed to be concave.We fix a point 2 e [0, a] which maximizes E f(s, x) 7rlh(x)-g(x) and assume that 0 < 2 < a.
Risk reduction: conditions on the cost and revenue functions.In the model, where f(s,) is defined by (57), the function () coincides with h(z) (see ( 51)).Therefore, by using Theorem 5.1, we immediately obtain the following result.
As we have already noticed, the inequality (0) 5 ( 1) implies (50).Thus, if the revenue function h(z) is concave, then the insurance mechanism reduces the uncertainty of the payoff in the sense of (50).If the function h(z) is not necessarily concave, but its squat e root is concave, then insurance reduces the vriance of the payoff.However, in this ce (50) may fail to hold.If h(z) is linear, then the pyoff (s) is non-random and its value is equal to E f(s, ).
We can see that the income (1) in case of success equals zero, while the income (0) in case of failure is strictly positive.Thus, insurance turns an unfavorable random outcome into a favorable one and vice versa.This situation is similar to that we saw in Section 4 (case 3).
One can give the following intuitive explanation of the bove facts.Suppose that the number m is large.Then so is the derivative g'() of the cost function g(:r) z m at the point z " (this derivative is precisely equal to m).Consequently, a small reduction of may significantly reduce possible losses.On the other hand, the derivative of the profit function f(1, x) at z is equal to 1. Thus, a small reduction of " cannot essentially reduce the profit in case of success.For this reason, if the decision-maker is risk-averse, then some decision z < may be much more preferable for him than the decision . .Therefore, a strong additional incentive may be needed to motivate the implementation of the plan '.The role of such an incentive is played by the large insurance compensation p(0) m-1, which is payed off in case of failure.Once the value of p(0) is large, so are the values of (0) and of Vat .O n the other hand, the random payoff /does not depend on m: we have r(0) = -1 and r/(1) 1.As a consequence of this, the random payoff turns out to be "much more variable" than , for large values of m.
Here, the variance of is equal to the variance of , however, (50) fails to hold.Indeed, if u(r) min{0, 1-r}, then Eu(,) -1/4 < 0 E u(rt).This remark shows that the inequality Var <Vat r/does not imply (50) not only for general random variables and ,, but also for those particular random variables ( and , which arise in our model. The case g(z) x.The remainder of the section will be devoted to the analysis of a special case of model (57).We shall assume that g(z) = z, z E [0, el, and so _j'h(x)-z, if s-l, I(s, z) -, if s=O.
(61) Here, is interpreted as the amount of money invested in the project.In case of success the project yields the revenue h(z).The net profit f(1, z) equals h(z)x.
In case of failure, no revenue is obtained and the amount invested is lost.Thus, we have f(0, x) = -.
The above assumptions permit us to use the results which have been established for the payoff function (57).In particular, formulas (59) and ( 60) can be used.
Reserves.Model (61) is a convenient vehicle for analyzing the following question.
Suppose the project manager is going to make the investment 2. Suppose he has a reserve amount, r, of money which he can use either for a direct compensation of possible losses or for insuring himself against those losses.What are the minimal necessary sizes, r. and r*, of the reserve r with insurance and without it?If the manager is not insured, then clearly his reserve r should not be less than 2, since the amount of losses in case of failure is .Consequently, we have that r* 2. If the manager wishes to be insured, then he has to pay the premium i6 (r0/;rl).The premium paid gives a right on the insurance compensation p(0) 2 + (r0/rl)', which covers the losses arising in case of failure.Thus we conclude 7r0_ The ratio r./r* r0/rl is small, provided that the probability 0 of failure is small, which is natural to assume in the present context.Consequently, the insurance mechanism under consideration makes it possible to reduce significantly times) the minimal necessary size of the individual reserve.
Also, it should be noted that there is an important distinction between the re- serves r, and r*.The former is renewable, while the latter is not.The profit f(1,) = h($)-, yielded by the project in case of success, exceeds r, i5 (see (65)) and is thus sufficient for renewing the reserve r,.The insurance com- pensation, paid off in case of failure, contains not only the amount but also the amount r, =/ (see (64)).Hence, the manager can get back the money he has spent for the premium and use it for the formation of his reserve.Thus the reserve r, can be renewed both in case of success and in case of failure.

Models with general state spaces
The problem of income stabilization.Consider the general model described in Section 2. Assume that the payoff function f(s, z) and the space X (C_ R) of decisions satisfy conditions (i)-(iv).Fix a decision E Z maximizing E f(s, z) over z E X. Throughout this section, we shll suppose that intX.Furthermore, it will be assumed that for each s S the function f(s, z) is differentiable with respect to z at the point z = .We shall not impose any restrictions on the measurable space S.
We use, basically, two different formalizations of the notion "less variable".To this end, we define appropriate preference relations in the space of random variables s e S. By definition, the purpose of the insurance mechanism (/5, p(.)) is to provide incen- tives for the realization of the plan (decision) which yields the maximum expected profit.If this plan, 5:, is unique, we can say that the insurance mechanism (, p(.)) eliminates the uncertainty in the choice of the plan.Indeed, with insurance the decision 2, and only it, becomes optimal for almost all s q S. Of course this does not mean that the uncertainty in the payoff yielded by the plan can always be eliminated.However, there are natural conditions under which the insurance mech- anism (/, p(.)) reduces this uncertainty, i.e., makes the random payoff more stable in the sense of one or another criterion.In the previous section, we described such conditions for nonlinear models with two states.In the present section, we obtain similar results for more general models.These results enable us to outline a uni- fied approach to the analysis of such aspects of insurance as support of effective decisions under risk and stabilization of the random income.
Two preference relations on the space of random variables.Denote by m the product measure t x g on S x S. Consider the space L of random variables (measurable functions) a" S --+ R1.Let a E L and/9 E L. To formalize the idea that a is less variable than/, we use two basic preference relations, 1 and , in the spce L. DEFINITION 6.1 We write fl a if I()-()1 I()-()! (7) m-Mmost everywhere on S x S (m-a.e.).DEFINITION 6.2 We define fl a if fl a and, in addition, we have 0 According to Definition 6.1, the random payoff a is preferred to the random payoff if almost every increment of a is not greater in its absolute value than the corresponding increment of .Definition 6.2 contains an additional requirement that the signs of the increments a(s)-a(s) and (s)-(s)can be opposite only for those (s., s2) which belong to a negligible subset of S x S. Note that both relations a and 2 a are defined without using expected utilities or any moments of the random variables a and .(We thus follow our general program" to avoid expected utilities and related concepts in the basic definitions.)Some conventional risk measures.The theorem below demonstrates links between the preference relations and and the traditional meures of risk involving concave utility functions and variances.THEOREM 6.1 Let a L and L be two random variables such that -< Ea, E<.By virtue of (b), if fl a, then every risk-averse individual will prefer the random payoff a to the random payoff ft.If a weaker relation fl a holds, we can only state that Vr a Vat $.
Observe that the expectations E u(a) and E u(fl), appearing in assertion (b), are well-defined.This is so, because for any concave function u R R we have u(r) u(O) + at, where a is some rel number.The vlues of E u(a) and E u(fl) may be finite or equal to -.
Assertion (b) will not be valid if we replace by .Indeed, take rndom variable a nd a concave function u satisfying E a 0 nd E u(a) < E u(-a).
Define $ -a.Then fl 5 a, while E u(a) < E u(fl).A similar example w considered in Section 5 (the ce m-3).
A necessary condition for .We postpone the proof of Theorem 6.1 till the end of this section.Our main goal is to examine conditions under which one or nother of the relations 5 and holds.Recall that (s) f(s, ), s e S, nd (s) f(s, ) /' (s, $)$ a.s. (see (66)).It will be convenient to ssume that the lt formula for (s) holds for all s 6 S, rather than a.s..This will not lead to a loss of generality.
Denote by '(sl, s2, ) the gradient of the function (Sl, s2,-) at the point .T his gradient exists, since f(s, x) is differentiable at x = by assumption.PROPOSITION 6.1 If _ , then we have (69)   almost everywhere with respect to the measure m.Proof: By using the definitions of and rt, we write (73) "More favorable" and "less favorable" states.In the course of further anal- ysis, we shall use the following definition.DEFINITION 6.3 Let sl S and s2 S. We say that the random outcome Sl is more favorable than s2, if f' (s, )" > f'(s2, (74) If (74) holds, we write s2 << s.
To understand the meaning of (74), assume for the moment that all vectors x in X are nonnegative and 0 X.Let us interpret coordinates zi, 1, 2,..., n, of the vector x = (z1,..., xn) X as intensities of realization of different parts (subprojects) of the project in question.For example, zi may specify the amount of money or resources invested in the ith subproject.Consider the function f0(,, ) f(,), where A is a real number.This function is defined, in particular, for all A's in a neighborhood of 1 (since E int X).The derivative f(s, ) h(s, ) g(z), (10o) where h(s, z) is the reee/unction (depending on the random parameter s) and g(z) is the cost function (independent of s).In accordance with the general sump- tions formulated at the beginning of the section, we suppose that the function f(s, z) satisfies (i)-(iii) and is differentiable with respect to x at z = 2 [ int X = (0, a)].Furthermore, the function h(s, z} is supposed to be smooth enough: we postulate the existence of all the partial derivatives of h(s, z) which are considered below.PROPOSITION (98 ,2 < O, seS, e [0,], then "<2 f.If for each s S the function h(s, z) is linear with respect to z, then (s) const (a.s.).
Proof: It follows from (101) that the derivative h'(s,) h (s,') is a strictly increasing function of s.Consequently, the relations s2 < sz, and (74) are equivalent.Therefore s2 << sl if and only if s2 < sl, and so (s (sh)).
Lemmas needed for proving Theorem 6.1.In the remainder of this section, we prove Theorem 6.1.The proof is based on several auxiliary facts.LEMMA 6.1 Let W be a closed subset of R d R d (d >_ 1) and (. S ---> R d a measurable vector function satisfying ((),(:)) e w for m-almost all (sl, s2) in S S. Then there exists a measurable vector function ' S --+ R d such that (s) '(s)/-a.e.
Proofs of the above two lemm are presented in the Appendix.
Proof: Let us prove the second assertion of the lemma; the first one is proved similarly.By virtue of Lemma 6.3, we may assume without loss of generality that relations (67) and (68) hold for all (81,82) in S x S, rather than m-a.e..For each r f(S), we define e(r) = a(s), where s is any element of fl-1 (r).In view of (67), a(s) does not depend on s E #-l(r), and so e(r) is well-defined for r #(S).

ROBUST INSURANCE MECHANISMS 115
The open set R \ cl (S) is a union of a finite or countable family of disjoint open intervals whose boundary points belong to cl f(S).For an infinite interval A = (-oo, c) or A = (c, +co) E, we set e(r) = e(c), r A.
Since -oo < E a = E < x, we obtain (110).From Lemma 6.2, it follows that E u() / u(e(r))(dr) >_ / u(r)(dr) E u() for any concave u(.).The model and the assumptions.In Section 3 we defined a Lagrange multi- plier p(-) associated with the information constraint (9) in the optimization problem (:P0).Using this Lagrange multiplier, we proved the existence of a robust insurance mechanism.Components of the vector function p(.) played the roles of compensa- tion prices in the robust insurance scheme.In the present section, we consider other applications of the Lagrange multipliers removing the information constraints.We use them for a study of the value of information in problems of decision making under risk.The main goal of the study is to develop a concept of shadow prices on information analogous to the well-known concept of shadow prices on resources (e.g.see Birchenhall and Grout [5]).
This section deals with the following model.We are given a space X of decisions and a state space S. It is assumed that the set X is contained in R n and satisfies conditions formulated in (iv) (see Section 2).The set S is finite: S {Sl,..., sg}.
There is a random parameter s taking values in S.This parameter takes each of the values sl,..., sN with strictly positive probabilities h,..., rg.We assume that a concave continuous functional F(z(.))isdefined on the set X = {z(-): z(s) E X, s e S} of decision functions (strategies) x(-) S --4 R'.The continuity of F(.) means that F(mk(.))--4 F(x(.)) if Zk(S) -z(s) for each s S. (Here, convergence of ink(S) for each s is equivalent to a.s.convergence.) We consider the optimization problems (P) and (71) described in Section 3. Re- call that the problem (:Pl) deals with the maximization of F(z(.)) over all functions z(-) X.In the problem (:P), we maximize F(z)over all constant functions in X, which can be identified with elements z in X.We fix a solution to (7) and denote this solution by .Clearly exists since F(z), z X, is continuous and X is compact (see condition (iv)).
In the previous part of the paper, we have basically dealt with functionals F(z(.)) representable in the form F(z(.)) Ef(s,z(s)), where f(s,m)is the payofffunction.
We have Et(.) = O, ( so that Et(-)" 0, and hence we conclude from (118) that < F(e)- = F(e) (120) for all (.) E X. Constraint (9) in the problem (:P0) does not allow the decision-maker to em- ploy any decision functions except constants.This means that the decision-maker ROBUST INSURANCE MECHANISMS 117 cannot use information about s.In this section we will consider relaxations of con- straint (9).These relaxations will correspond to certain (limited) possibilities of using information about s.
Decisions and their admissible corrections.To explain the idea of our ap- proach, consider the model of realization of a risky project described in Sections 1 and 2. In that model, the project has to be realized during the period [to, tl].The manager takes a decision x E X at time to.Until the end of the project period, he does not receive any information about s E S which could be used for changing the initial plan.If the goal of the decision-maker is to maximize the functional F(.), then his optimal decisions are solutions to the problem (:P0).
Now suppose that there is a source of information which makes it possible to learn the exact value of s S before the end of the project period [to, t l].Suppose the knowledge of s can be used for making a correction of the initial plan.This means that the initial decision x X can be replaced by a decision function z(s) -t-h(s), where h(s) is the correction of the decision x at state s.For example, if the manager learns that the outcome of the project will be unfavorable, he may stop payments (or supply) in order to reduce future losses.On the other hand, if the outcome of the project turns out to be favorable, the manager may increase investments to get a greater reward.The earlier the decision-maker receives information about s, the more freedom for changing the initial decision he has, and so the larger is the class of admissible corrections h(.).Based on the above considerations, we shall characterize a source of irformator, I, in terms of the class 7/ = I of all admissible corrections h(.) S --+ R , this source of information enables one to use.To the class 7/of admissible corrections, there corresponds the class = e x: = + e x, e 7t} (121) of admissible decision functions.According to (121), a function x(-) X belongs to 3)(7/) if and only if x(-) is representable in the form z(.) = z + h(.), where x X is the initial decision and h(.) E 7/ is the correction.The possibility of using a certain class 32 = Y(7/) of decision functions (strategies) z(.) S --+ X (in addition to decisions z X independent of s) will be regarded as a relaxation of the information constraint.Indeed, under the information constraint the only admissible correction is the zero correction.
The class of admissible corrections: some basic properties.Later on we will discuss the specification of the class 7/= 7/I of admissible corrections in more detail.In this stage of exposition it is important to realize that without loss of generality one may restrict the attention to corrections h(.) belonging to the set , defined as ={h(.)el:zeX,z + h(s) X for all s e S}, where : is the finite-dimensional linear space of all vector functions h(.) S --R n Indeed, if h(.)G then z + h(.)X for all z e Z so that such an h(.) gives no contribution to classes as Y(Tt) defined in (121).divisible portions of information.If S is finite (which has been assumed in this section), then there is only a finite number of partitions of S. Consequently, the "amount of information" can vary by discrete portions only.Furthermore, if S consists of two points (success and failure), then there are only two partitions of S: the trivial one S, }, and the partition of S into points.In other words, there is either no information or complete information about s; the conventional approach does not enable us to consider any intermediate cases.
It would be of interest to extend the results of this section to more general models.In particular, one can try to analyze the concept of shadow prices on information in the framework of dynamic (multi-stage) problems of decision making under un- certainty.Another interesting field for possible applications and generalizations of the methods developed is the theory of teams (Marschak and Radner [17]).
Ea E$, then E u(a) Eu() for any concave function u" R R .

o 7 .
The shadow prices of information constraints