2. UTILITY THEORY
2.
29
Utility Theory
2.1. The Expected Utility Hypothesis
We consider an agent carrying a risk, for example a person who is looking for insurance, an insurance company looking for reinsurance, a financial engineer with
a certain portfolio. Alternatively, we could consider an agent who has to decide
whether to take over a risk, as for example an insurance or a reinsurance company.
We consider a simple one-period model. At the beginning of the period the agent
has the wealth w, at the end of the period he has the wealth w − X, where X is the
loss occurred from the risk. For simplicity we assume X ≥ 0.
The agent is now offered an insurance contract. An insurance contract is
specified by a compensation function r : IR+ → IR+ . The insurer covers the
amount r(X), the insured covers X − r(X). For the compensation of the risk taken
over by the insurer, the insured has to pay the premium π. Let us first consider
the properties r(x) should have. Because the insured looks for security, the function
r(x) should be non-negative. Otherwise, the insured may take over a larger risk
(for a possibly negative premium). On the other side, there should be no overcompensation, i.e. 0 ≤ r(x) ≤ x. Otherwise, the insured could make a gain from a
loss. Let s(x) = x − r(x) be the self-insurance function, the amount covered by
the insured. Sometimes s(x) is also called franchise.
The insured has now the problem to find the appropriate insurance cover. If
he uses the policy (π, r(x)) he will have the wealth Y = w − π − s(X) after the
period. A first idea would be to consider the expected wealth IIE[Y ] and to choose
the policy with the largest expected wealth. Because some administration costs
will be included in the premium, the optimal decision will then always be not to
insure the risk. The agent wants to reduce his risk, and therefore is interested in
buying insurance. He therefore would be willing to pay a premium π such that
IIE[Y ] < IIE[w − X], provided the risk is reduced. A second idea would be to compare
the variances. Let Y1 and Y2 be two wealths corresponding to different insurance
treaties. We prefer the first insurance form if Var[Y1 ] < Var[Y2 ]. This will only
make sense if IIE[Y1 ] = IIE[Y2 ]. The disadvantage of the above approach is, that a
loss IIE[X] − x will be considered as bad as a loss IIE[X] + x. Of course, the insured
prefers a smaller to a larger loss.
The above discussion inspires the following concept. The agent gives a value to
each possible wealth; i.e., he has a function u : I → IR, called utility function,
30
2. UTILITY THEORY
giving a value to each possible wealth. Here I is the interval of possible wealths
attainable by any insurance under consideration. I is assumed not to be a singleton.
His criterion will now be expected utility; i.e., he will prefer insurance form one
if IIE[u(Y1 )] > IIE[u(Y2 )].
We now try to find properties a utility function should have. The agent will of
course prefer larger wealth. Thus u(x) must be strictly increasing. Moreover, the
agent is risk averse, i.e. he will weight a loss higher for small wealth than for large
wealth. Mathematically we can express this in the following way. For any h > 0
the function y 7→ u(y + h) − u(y) is strictly decreasing. Note that if u(y) is a utility
function fulfilling the above conditions, then also ũ(y) = a + bu(y) for a ∈ IR and
b > 0 fulfils the conditions. Moreover, ũ(y) and u(y) will lead to the same decisions.
Let us consider some possible utility functions.
• Quadratic utility
u(y) = y −
y2
,
2c
y ≤ c, c > 0.
(2.1a)
y ∈ IR, c > 0.
(2.1b)
• Exponential utility
u(y) = −e−cy ,
• Logarithmic utility
u(y) = log(c + y),
y > −c.
(2.1c)
• HARA utility
u(y) =
(y + c)α
α
or
u(y) = −
(c − y)α
α
y > −c, 0 < α < 1,
y < c, α > 1.
(2.1d)
(2.1e)
The second condition for a utility function looks a little bit complicated. We therefore want to find a nicer equivalent condition.
Theorem 2.1. A strictly increasing function u(y) is a utility function if and only
if it is strictly concave.
Proof.
Suppose u(y) is a utility function. Let x < z. Then
u(x + 21 (z − x)) − u(x) > u(z) − u(x + 21 (z − x))
implying
u( 12 (x + z)) > 12 (u(x) + u(z)) .
(2.2)
2. UTILITY THEORY
31
We fix now x and z > x. Define the function v : [0, 1] → [0, 1],
α 7→ v(α) =
u((1 − α)x + αz) − u(x)
.
u(z) − u(x)
Note that u(y) concave is equivalent to v(α) ≥ α for α ∈ (0, 1). By (2.2) v( 21 ) > 21 .
Let n ∈ IIN and let αn,j = j2−n . Assume v(αn,j ) > αn,j for 1 ≤ j ≤ 2n − 1. Then
clearly v(αn+1,2j ) = v(αn,j ) > αn,j = αn+1,2j . By (2.2) and 12 (αn,j + αn,j+1 ) =
αn+1,2j+1 ,
v(αn+1,2j+1 ) > 21 (v(αj,n ) + v(αn,j+1 )) > 21 (αn,j + αn,j+1 ) = αn+1,2j+1 .
Let now α be arbitrary and let jn such that αn,jn ≤ α < αn,jn +1 , αn = αn,jn . Then
v(α) ≥ v(αn ) > αn .
Because n is arbitrary we have v(α) ≥ α. This implies that u(y) is concave. Let
now α < 21 . Then
u((1 − α)x + αz) = u((1 − 2α)x + 2α 12 (x + z)) ≥ (1 − 2α)u(x) + 2αu( 21 (x + z))
> (1 − 2α)u(x) + 2α 21 (u(x) + u(z)) = (1 − α)u(x) + αu(z) .
A similar argument applies if α > 21 . The converse statement is left as an exercise.
Next we would like to have a measure, how much an agent likes the risk. Assume
that u(y) is twice differentiable. Intuitively, the more concave a utility function is,
the more it is sensible to risk. So a risk averse agent will have a utility function
with −u00 (y) large. But −u00 (y) is not a good measure for this purpose. Let ũ(y) =
a + bu(y). Then −ũ00 (y) = −bu00 (y) would give a different risk aversion, even though
it leads to the same decisions. In order to get rid of b define the risk aversion
function
u00 (y)
d
a(y) = − 0
= − ln u0 (y) .
u (y)
dy
Intuitively, we would assume that a wealthy person can take more risk than a poor
person. This would mean that a(y) should be decreasing. For our examples we have
that (2.1a) and (2.1e) give increasing risk aversion, (2.1c) and (2.1d) give decreasing
risk aversion, and (2.1b) gives constant risk aversion.
32
2. UTILITY THEORY
2.2. The Zero Utility Premium
Let us consider the problem that an agent has to decide whether to insure his risk
or not. Let (π, r(x)) be an insurance treaty and assume that it is a non-trivial
insurance, i.e. Var[r(X)] > 0. If the agent takes the insurance he will have the final
wealth w − π − s(X), if he does not take the insurance his final wealth will be w − X.
Considering his expected utility, the agent will take the insurance if and only if
IIE[u(w − π − s(X))] ≥ IIE[u(w − X)] .
If equality holds, the agent is indifferent. So taking the insurance is possible. The left
hand side of the above inequality is a continuous and strictly decreasing function of π.
The largest value that can be attained is for π = 0, IIE[u(w − s(X))] > IIE[u(w − X)],
the lowest possible value is difficult to obtain, because it depends on the choice of
I. Let us assume that the lower bound is smaller or equal to IIE[u(w − X)]. Then
there exists a unique value πr (w) such that
IIE[u(w − πr − s(X))] = IIE[u(w − X)] .
The premium πr is the largest premium the agent is willing to pay for the insurance
treaty r(x). We call πr the zero utility premium. In many situations the zero
utility premium πr will be larger then the “fair premium” IIE[r(X)].
Theorem 2.2. If both r(x) and s(x) are increasing and Var[r(X)] > 0 then
πr (w) > IIE[r(X)].
Proof. From the definition of a strictly concave function it follows that the function v(y), x 7→ v(x) = u(w − x) is a strictly concave function. Define the functions g1 (x) = s(x) + IIE[r(X)] and g2 (x) = x. These two functions are increasing.
The function g2 (x) − g1 (x) = r(x) − IIE[r(X)] is increasing. Thus there exists x0
such that g1 (x) ≥ g2 (x) for x < x0 and g1 (x) ≤ g2 (x) for x > x0 . Note that
IIE[g2 (x) − g1 (x)] = 0. Thus the conditions of Corollary G.7 are fulfilled and
IIE[u(w − IIE[r(X)] − s(X))] = IIE[v(g1 (X))] > IIE[v(g2 (X))]
= IIE[u(w − X)] = IIE[u(w − πr − s(X))] .
Because π 7→ IIE[u(w − π − s(X))] is a strictly decreasing function it follows that
πr > IIE[r(X)].
Assume now that u is twice continuously differentiable. Then we find the following result.
2. UTILITY THEORY
33
Theorem 2.3. Let r(x) = x. If the risk aversion function is decreasing (increasing), then the zero utility premium π(w) is decreasing (increasing).
Proof.
Consider the function v(y) = u0 (u−1 (y)). Taking the derivative yields
v 0 (y) =
u00 (u−1 (y))
= −a(u−1 (y)) .
0
−1
u (u (y))
Because y 7→ u−1 (y) is increasing, this gives that v 0 (y) is increasing (decreasing).
Thus v(y) is convex (concave).
We have
π(w) = w − u−1 (IIE[u(w − X)]) and π 0 (w) = 1 −
IIE[u0 (w − X)]
.
u0 (u−1 (IIE[u(w − X)]))
Let now a(y) be decreasing (increasing). By Jensen’s inequality we have
u0 (u−1 (IIE[u(w − X)])) ≤ (≥) IIE[u0 (u−1 (u(w − X)))] = IIE[u0 (w − X)]
and therefore π 0 (w) ≤ (≥) 0.
It is not always desirable that the decision depends on the initial wealth. We
have the following result.
Lemma 2.4. Let u(x) be a utility function. Then the following are equivalent:
i) u(x) is exponential, i.e. u(x) = −Ae−cx + B for some A, c > 0, B ∈ IR.
ii) For all losses X and all compensation functions r(x), the zero utility premium
πr does not depend on the initial wealth w.
iii) For all losses X and for full reinsurance r(x) = x, the zero utility premium πr
does not depend on the initial wealth w.
Proof.
This follows directly from Lemma 1.8.
2.3. Optimal Insurance
Let us now consider an agent carrying several risks X1 , X2 , . . . , Xn . The total risk is
then X = X1 + · · · + Xn . We consider here a compensation function r(X1 , . . . , Xn )
and a self-insurance function s(X1 , . . . , Xn ) = X − r(X1 , . . . , Xn ). We say an
insurance treaty is global if r(X1 , . . . , Xn ) depends on X only, or equivalently,
s(X1 , . . . , Xn ) depends on X only. Otherwise, we call an insurance treaty local. We
next prove that, in some sense, the global treaties are optimal for the agent.
34
2. UTILITY THEORY
Theorem 2.5. (Pesonen-Ohlin) If the premium depends on the pure net premium only, then to every local insurance treaty there exists a global treaty with the
same premium but a higher expected utility.
Proof. Let (π, r(x1 , . . . , xn )) be a local treaty. Consider the compensation function R(x) = IIE[r(X1 , . . . , Xn ) | X = x]. If the premium depends on the pure net
premium only, then clearly R(X) is sold for the same premium, because
IIE[R(X)] = IIE[IIE[r(X1 , . . . , Xn ) | X]] = IIE[r(X1 , . . . , Xn )] .
By Jensen’s inequality
IIE[u(w − π − s(X1 , . . . , Xn ))] = IIE[IIE[u(w − π − s(X1 , . . . , Xn )) | X]]
< IIE[u(IIE[w − π − s(X1 , . . . , Xn ) | X])] = IIE[u(w − π − (X − R(X)))] .
The strict inequality follows from the fact that s(X1 , . . . , Xn ) 6= S(X), because
otherwise the insurance treaty would be global, and from the strict concavity. Remark. Note that the result does not say that the global treaty constructed is
a good treaty. It only says, that for the investigation of good treaties we can restrict
to global treaties.
Let us now find the optimal insurance treaty for the insured.
Theorem 2.6. (Arrow-Ohlin) Assume the premium depends on the pure net
premium only and let (π, r(x)) be an insurance treaty with IIE[r(X)] > 0. Then
there exists a unique deductible b > 0 such that the excess of loss insurance
rb (x) = (x − b)+ has the same premium. Moreover, if IIP[r(X) 6= rb (X)] > 0 then
the treaty (π, rb (x)) yields a higher utility.
Proof. The function (x − b)+ is continuous and decreasing in b. We have IIE[(X −
0)+ ] ≥ IIE[r(X)] > 0 = limb→∞ IIE[(X − b)+ ]. Thus there exists a unique b > 0 such
that IIE[(X − b)+ ] = IIE[r(X)]. Thus rb (x) yields the same premium. Assume now
IIP[r(X) 6= rb (X)] > 0. Let sb (x) = x − rb (x). For y < b we have
IIP[sb (X) ≤ y] = IIP[X ≤ y] ≤ IIP[s(X) ≤ y] ,
and for y > b
IIP[sb (X) ≤ y] = 1 ≥ IIP[s(X) ≤ y] .
2. UTILITY THEORY
35
Note that v(x) = u(w − π − x) is a strictly concave function. Thus Ohlin’s lemma
yields
IIE[u(w − π − sb (X))] = IIE[v(sb (X))] > IIE[v(s(X))] = IIE[u(w − π − s(X))] .
2.4. The Position of the Insurer
We now consider the situation from the point of view of an insurer. The discussion
how to take decisions does also apply to the insurer. Assume therefore that the
insurer has the utility function ū(x). Denote the initial wealth of the insurer by w .
For an insurance treaty (π, r(x)) the expected utility of the insurer becomes
IIE[ū(w + π − r(X))] .
The zero utility premium π r (w ) of the insurer is the the solution to
IIE[ū(w + π r (w ) − r(X))] = ū(w ) .
The insurer will take over the risk if π ≥ π r (w ). By Jensen’s inequality we have
ū(w ) = IIE[ū(w + π r (w ) − r(X))] < ū(w + π r (w ) − IIE[r(X)])
which yields π r (w ) − IIE[r(X)] > 0. A insurance contract can thus be signed by both
parties if and only if π r (w ) ≤ π ≤ πr (w). It follows as in Theorem 2.5 (replacing
s(x1 , . . . , xn ) by r(x1 , . . . , xn )) that the insurer prefers global treaties to local ones.
From Theorem 2.6 (replacing s(x) by r(x)) it follows that a first risk deductible is
optimal for the insurer. Thus the interests of insurer and insured contradict.
Let us now consider insurance forms that take care of the interests of the insured.
The insured would like that the larger the loss the larger should be the fraction
covered by the insurance company. We call a compensation function r(x) a Vajda
compensation function if r(x)/x is an increasing function of x. Because rb (x) =
(x − b)+ is a Vajda compensation function it is clear how the insured would choose
r(x). Let us now consider what is optimal for the insurer.
Theorem 2.7. Suppose the premium depends on the pure net premium only. For
any Vajda insurance treaty (π, r(x)) with IIE[r(X)] > 0 there exists a proportional
insurance treaty rk (x) = (1−k)x with the same premium. Moreover, if IIP[r(X) 6=
rk (X)] > 0 then (π, rk (x)) yields a higher utility for the insurer.
36
2. UTILITY THEORY
Proof. The function [0, 1] → IR : k 7→ IIE[(1 − k)X] is decreasing and has its
image in the interval [0, IIE[X]]. Because 0 < IIE[r(X)] ≤ IIE[X] there is a unique k
such that IIE[rk (X)] = IIE[r(X)]. These two treaties have then the same premium.
Suppose now that IIP[r(X) 6= rk (X)] > 0. Note that r(x) and rk (x) are increasing.
The difference
r(x) − rk (x) r(x)
=
− (1 − k)
x
x
is an increasing function in x. Thus there is x0 such that r(x) ≤ rk (x) for x < x0
and r(x) ≥ rk (x) for x > x0 . The function v(x) = ū(w + π − x) is strictly concave.
By Corollary G.7 we have
IIE[ū(w + π − rk (X))] = IIE[v(rk (X))] > IIE[v(r(X))] = IIE[ū(w + π − rk (X))] .
This proves the result.
As from the point of view of the insured it turns out that the zero utility premium
is only independent of the initial wealth if the utility is exponential.
Proposition 2.8. Let ū(x) be a utility function. Then the following are equivalent:
i) ū(x) is exponential, i.e. ū(x) = −Ae−cx + B for some A, c > 0, B ∈ IR.
ii) For all losses X and all compensation functions r(x), the zero utility premium
π r does not depend on the initial wealth w .
Proof.
The result follows analogously to the proof of Proposition 2.4.
2.5. Pareto-Optimal Risk Exchanges
Consider now a market with n agents, i.e. insured, insurance companies, reinsurers.
In the market there are the risks X = (X1 , . . . , Xn )> . Some of the risks may be
zero, if the corresponding agent does not initially carry a risk. We allow here also
for financial risks (investment). If Xi is positive, we consider it as a loss, if it is
negative, we consider it as a gain. Making contracts, the agents redistribute the risk
X. Formally, a risk exchange is a function
f = (f1 , . . . , fn )> : IRn → IRn ,
2. UTILITY THEORY
such that
37
n
X
i=1
fi (X) =
n
X
Xi .
(2.3)
i=1
The latter condition assures that the whole risk is covered. A risk exchange means
that agent i covers fi (X). This is a generalisation of the insurance market we had
considered before.
Now each of the agents has an initial wealth wi and a utility function ui (x).
Under the risk exchange f the expected terminal utility of agent i becomes
Vi (f ) = IIE[ui (wi − fi (X))] .
As we saw before, the interests of the agents will contradict. Thus it will in general
not be possible to find f such that each agent gets an optimal utility. However, an
agent may agree to a risk exchange f if Vi (f ) ≥ IIE[ui (wi − Xi )]. Hence, which risk
exchange will be chosen depends upon negotiations. However, the agents will agree
not to discuss certain risk exchanges.
Definition 2.9. A risk exchange f is called Pareto-optimal if for any risk exchange f˜ with
Vi (f ) ≤ Vi (f˜) ∀i
it follows that
Vi (f ) = Vi (f˜) ∀i .
For a risk exchange f that is not Pareto optimal we would have a risk exchange f˜
that is at least as good for all agents, but better for at least one of them. Of course,
it is not excluded that Vi (f ) < Vi (f˜) for some i. If the latter is the case then there
must be j such that Vj (f ) > Vj (f˜).
For the negotiations, the agents can in principle restrict to Pareto-optimal risk
exchanges. Of course, in reality, the agents do not know the utility functions of the
others. So the concept of Pareto optimality is more a theoretical way to consider
the market.
It seems quite difficult to decide whether a certain risk exchange is Pareto-optimal
or not. It turns out that a simple criteria does the job.
Theorem 2.10. (Borch/du Mouchel) Assume that the utility functions ui (x)
are differentiable. A risk exchange f is Pareto-optimal if and only if there exist
numbers θi such that (almost surely)
u0i (wi − fi (X)) = θi u01 (w1 − f1 (X)) .
(2.4)
38
2. UTILITY THEORY
Remark. Note that necessarily θ1 = 1 and θi > 0 because u0i (x) > 0. Note also
that we could make the comparisons with agent j instead of agent 1. Then we just
need to divide the θi by θj , for example θ1 would be replaced by θj−1 .
Proof. Assume that θi exist such that (2.4) is fulfilled. Let f˜ be a risk exchange
such that Vi (f˜) ≥ Vi (f ) for all i. By concavity
ui (wi − f˜i (X)) ≤ ui (wi − fi (X)) + u0i (wi − fi (X))(fi (X) − f˜i (X)) .
This yields
ui (wi − f˜i (X)) − ui (wi − fi (X))
≤ u01 (w1 − f1 (X))(fi (X) − f˜i (X)) .
θi
P
P
P
Using ni=1 fi (X) = ni=1 Xi = ni=1 f˜i (X) we find by summing over all i,
n
X
ui (wi − f˜i (X)) − ui (wi − fi (X))
θi
i=1
≤ 0.
Taking expectations yields
n
X
Vi (f˜) − Vi (f )
i=1
θi
≤ 0.
From the assumption Vi (f˜) ≥ Vi (f ) it follows that Vi (f˜) = Vi (f ). Thus f is
Pareto-optimal.
Assume now that (2.4) does not hold. By renumbering, we can assume that
there is no constant θ2 such that (2.4) holds for i = 2. Let
θ=
IIE[u01 (w1 − f1 (X))u02 (w2 − f2 (X))]
IIE[(u01 (w1 − f1 (X)))2 ]
and W = u02 (w2 −f2 (X))−θu01 (w1 −f1 (X)). We have then IIE[W u01 (w1 −f1 (X))] = 0
and by the assumption IIE[W 2 ] > 0. Let now ε > 0, δ = 21 IIE[W 2 ]/IIE[u02 (w2 −
f2 (X))] > 0 and f˜1 (X) = f1 (X) − (δ − W )ε, f˜2 (X) = f2 (X) + (δ − W )ε, f˜i (X) =
fi (X) for all i > 2. Then f˜ is a risk exchange. Consider now
V1 (f˜) − V1 (f )
− IIE[u01 (w1 − f1 (X))]δ
ε
h
u (w − f˜ (X)) − u (w − f (X))
i
1
1
1
1
1
1
= IIE (δ − W )
− u01 (w1 − f1 (X)) ,
(δ − W )ε
2. UTILITY THEORY
39
where we used IIE[W u01 (w1 − f1 (X))] = 0. The right-hand side tends to zero as ε
tends to zero. Thus V1 (f˜) > V1 (f ) for ε small enough. We also have
V2 (f˜) − V2 (f )
− IIE[u02 (w2 − f2 (X))(W − δ)]
ε
h
u (w − f˜ (X)) − u (w − f (X))
i
2
2
2
2
2
2
0
= IIE (W − δ)
− u2 (w2 − f2 (X)) ,
(W − δ)ε
Also here the right-hand side tends to zero. We have
IIE[u02 (w2 − f2 (X))(W − δ)] = IIE[(W + θu01 (w1 − f1 (X)))W ] − δIIE[u02 (w2 − f2 (X))]
= IIE[W 2 ] − δIIE[u02 (w2 − f2 (X))] = 21 IIE[W 2 ] > 0 .
Thus also V2 (f˜) > V2 (f ) for ε small enough. This shows that f is not Paretooptimal.
In order to find the Pareto-optimal risk exchanges one has to solve the equation
(2.4) subject to the constraint (2.3).
If a risk exchange is chosen, the quantity fi (0, . . . , 0) is the amount agent i has
to pay (obtains if negative) if no losses occur. Thus fi (0, . . . , 0) must be interpreted
as a premium.
Note that for Pareto-optimality only the support of the distribution of X, not
the distribution itself, does have an influence. Hence the Pareto-optimal solution
can be found without investigating the risk distribution, nor the dependencies. It
P
also shows that any Pareto optimal solution will be a function of ni=1 Xi only.
We now solve the problem explicitly. Because u0i (x) is strictly decreasing, it is
invertible and its inverse (u0i )−1 (y) is strictly decreasing. Moreover, if u0i (x) is continuous (as it is under the conditions of Theorem 2.10), then (u0i )−1 (y) is continuous,
too. Choose θi > 0, θ1 = 1. Then (2.4) yields
wi − fi (X) = (u0i )−1 (θi u01 (w1 − f1 (X))) .
Summing over i gives by use of (2.3)
n
X
i=1
wi −
n
X
Xi =
i=1
The function
g(y) =
n
X
(u0i )−1 (θi u01 (w1 − f1 (X))) .
i=1
n
X
i=1
(u0i )−1 (θi u01 (y))
40
2. UTILITY THEORY
is then strictly increasing and therefore invertible. If all the u0i (x) are continuous,
then also g(y) is continuous and therefore g −1 (x) is continuous, too. This yields
f1 (X) = w1 − g
−1
n
X
wi −
i=1
n
X
Xi .
i=1
For i arbitrary we then obtain
fi (X) = wi −
(u0i )−1
θi u01
g
−1
n
X
i=1
wi −
n
X
Xi
.
i=1
Thus we have found the following result.
Theorem 2.11. A Pareto-optimal risk exchange is a pool, that is each individual
agent contributes a share that depends on the total loss of the group only. Moreover,
each agent must cover a genuine part of any increase of the total loss of the group.
If all utility functions are differentiable, then the individual shares are continuous
functions of the total loss.
Remark. Consider the situation insured-insurer, i.e. n = 2 and a risk of the form
(X, 0)> , where X ≥ 0. For certain choices of θ2 it is possible that f1 (0, 0) < 0, i.e. the
insurer has to pay a premium to take over the risk. This shows that not all possible
choices of θi will be realistic. In fact, the risk exchange has to be chosen such that the
expected utility of each agent is increased, i.e. IIE[ui (wi − fi (X))] ≥ IIE[ui (wi − Xi )]
in order that agent i will be willing to participate.
Bibliographical Remarks
The results presented here can be found in [80].