The name is absent

Incremental Risk Vulnerability

so that

q 1∣∆( y 1)| = q 2∆( y 2)

Now r(w) can be rewritten from (6) as

E ʃ u'(W) -u"(W)
^y ∣Ey[u'(W)] u'(W)
Ey {Eu-WWÎ^r ( ^w ’}

Hence, ^(w) is the expected value of the coefficient of absolute risk aversion, using the
risk-neutral probabilities given by the respective probabilities multiplied by the ratio of
the marginal utility to the expected marginal utility. Thus, r(w) is a convex combination
of the coefficients of absolute risk aversion at the different values of y . For the three-
point distribution being considered, ^(w) is a convex combination of r(W0), r(W₁), and
r(W₂). Suppose that y0 = 0. Then q0 → 1 is feasible. Hence, as q0 → 1, ^(w) → r(W₀).
Therefore, in condition (9) we replace ^(w) by r(W₀). Since W0 can take any value in
the range [W1,W2], f (w, y, s) must have the required sign for every value of r(W0), where
W1 ≤ W0 ≤ W2. Thus, since q1 ∣∆(y1)∣ > 0, the condition as stated in Proposition 1 must
hold. As y ∈ (y, y), W2 - W1 < y - y.

Sufficiency

To establish sufficiency we use a method similar to that used by Pratt and Zeckhauser
(1987) and Gollier and Pratt (1996).

a) We first show

u'"(W2) - u"'(W1) < -r(W) [u"(W2) - u¹¹(W1)] , ∀ W1 ≤ W ≤ W2
=⇒ f (w, y, s) > 0, ∀ (w, y, s)

We need to show that f (w, y, s) > 0, for all non-degenerate probability distributions of y.
Hence, we need to prove that the minimum value of f (w, y, s) over all possible probability
distributions {qi}, with E(∆(y)) = 0, must be positive. In a manner similar to Gollier
and Pratt (1996), this can be formulated as a mathematical programming problem, where
f (w, y, s) is minimized, subject to the constraints that all qi are non-negative and sum

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