Donations under Ambiguity

Similarly to Section 4.2.1.1, the survival rate for both cancer types is given by ps(d) =

p0

s+(1 p0s) (d),wherep0shas some unique value in the interval (0,1).In contrast to Section

4.2.1.1 where pa also had a unique value, the decision-maker’s beliefs about the likelihood

that cancerlsis induced by a specific, avoidable, lifestyle are given by theprobability interval

⇥

pa,p¯a

⇤

, where 0  pa  p¯a  1. pa and ¯pa are called minimal and maximal probabilities,

respectively. As before, the corresponding probability for cancer hr is assumed to be equal to zero.

The elements of ⇥pa,p¯a

⇤

can be thought of as representing at least two factors: the decision-maker’s information on the possible probabilities and his or her degree of confidence in the existing theories surrounding these probabilities.7 _{So, for example, if there are several}

competing hypotheses about the likelihood that cancer ls is induced by a specific lifestyle, but the donor is convinced that only one is truly valid, then ⇥pa,p¯a

⇤

would be a singleton. In this case, the decision-maker faces pure risk. In contrast, complete ambiguity or pure uncertainty is characterized by ⇥pa,p¯a

⇤

= [0,1]. This represents situation where the donor has no information on the current likelihood that cancer ls is induced by a specific lifestyle other than that it falls somewhere in [0,1]. We will call the di↵erence between the maximal and minimal probabilities a degree of ambiguity and denote it by ! ⌘p¯a pa.

7_{Gajdos et al. (2004) provide a complementary interpretation of the set of probabilities. The decision-}

maker in their model maximizes the minimum expected utility computed with respect to a subset of the set of initially given priors. The extent to which the set of initially given priors is reduced is a measure of aversion to information imprecision.

The donor has ↵ MMEU preferences Wi(d) =↵ min pa2[pa,p¯a] EU_ihr+EU_ils(pa) (4.10) + (1 ↵) max pa2[pa,p¯a] EU_ihr+EU_ils(pa) ,

where↵ ₂[0,1] is the parameter characterizing the donor’s attitude to ambiguity,i_2{n, ch_} reflects whether the donor is a choice egalitarian or a non-choice egalitarian, and EUhr

i and

EUls

i (pa) are given by (4.2) fori=n and (4.4) fori=ch.↵ MMEU preference structure is

a generalization of Arrow-Hurwicz criterion (Hurwicz, 1952, Arrow and Hurwicz, 1972) and it reduces to an expected utility preference functional when⇥pa,p¯a

⇤

is a singleton.

↵ MMEU preference structure is also a natural generalization of the maximin and

maximax decision rules (Gilboa and Schmeidler, 1989). When↵ = 1,↵ MMEU preferences

havemaximin expected utility form (Gilboa and Schmeidler, 1989) which corresponds tocom-

plete ambiguity aversion (also called complete pessimism) on the donor’s part. An MMEU

donor is completely pessimistic in the sense that, when evaluating stochastic outcomes, he or she always uses the probability distribution that yields the lowest possible expected utility over⇥pa,p¯a

⇤

. In our case, an MMEU donor will use the smallest possible probabilitypa that

cancer lsis induced by lifestyle.

In contrast, ↵ = 0 corresponds to a donor with maximax expected utility preferences and reflects a situation where the donor is completely ambiguity tolerant. A donor with

↵= 0 focuses all attention on themost optimistic probability distribution, which in our case is equal to ¯pa.

Given that ↵ MMEU utility is a weighted linear functional of the most pessimistic and most optimistic scenarios, it is natural to call ↵ a measure of ambiguity aversion or a

more pessimistic).8 _{A donor with ↵ > 0.5 is said to be ambiguity averse while a donor}

with ↵ < 0.5 is said to be ambiguity loving. Note also that a donor with ↵ = 0.5 is not ambiguity neutral. In the present model, a donor is ambiguity neutral if and only if the set of probabilities ⇥pa,p¯a

⇤

is a singleton.

The utilities assigned by the choice egalitarian and non-choice egalitarian donors to the two cancer patients are identical to the corresponding utilities in Section 4.2.1.1. Under this assumption and using (4.1), (4.2) and (4.10), we can write the non-choice egalitarian donor’s objective function as

Wn(d) = ps(d) +ps(100 d),

while the choice egalitarian donor’s objective as

Wch(d) = ps(d) +ps(100 d) + ↵pa+ (1 ↵) ¯pa (1 ps(100 d)).

The latter expression reveals that the choice-egalitarian donor with ↵ MMEU preference and beliefs ⇥pa,p¯a

⇤

behaves similarly to the choice-egalitarian donor with expected utility preferences and beliefs given by the singleton probability ↵pa+ (1 ↵) ¯pa .It also follows

immediately from the objective functions above that the non-choice egalitarian donor’s op- timal donation is unchanged from Section 4.2.1.1 and is given by (4.3). In contrast, the condition characterizing the choice egalitarian donor’s optimal donation changes from (4.5) to:

1 p0_s ⇥ 0(d) 1 ↵pa+ (1 ↵) ¯pa 0(100 d)

⇤

= 0. (4.11)

Using the expression for Wn(d) and (4.11), we obtain

8_{Note that “more ambiguity averse” is a comparative rather than absolute notion. Thus, a more ambiguity}

Proposition 3 If the donor is non-choice egalitarian then her donation to the treatment of the lifestyle-related cancer will be una↵ected by her ambiguity aversion and her beliefs ⇥pa,p¯a

⇤

that cancer ls is lifestyle-related. Conversely, if the donor is choice egalitarian then

(a) a more ambiguity averse donor will donate more to the treatment of the lifestyle-related

cancer;

(b) an increase in the minimal likelihood of the lifestyle-related cancer, holding its degree of ambiguity constant, dpa=d¯pa>0 will result in a decrease in the donation to the treatment

of the lifestyle-related cancer.

(c)an increase in the ambiguity of the lifestyle-related cancer, holding its minimal likelihood constant, dp¯a > 0 = dpa will result in a decrease in the donation to the treatment of the

lifestyle-related cancer.

The intuition behind Proposition 3 can be grasped immediately by examining the expressions for the two donor types’ objective functions. While the attitude and perception of ambiguity don’t a↵ect the non-choice egalitarian donor, these behavioral traits enter the choice egalitarian donor’s objectiveWch(d) as the sum ↵pa+ (1 ↵) ¯pa of the minimal and

maximal probabilities weighted by the respective ambiguity attitudes. This term is decreas- ing in the degree of ambiguity aversion and increasing inpaand ¯pa.Recall also that the larger

the magnitude of ↵pa+ (1 ↵) ¯pa the larger the utility assigned by the choice egalitarian

to the adverse outcome uls_{(M) associated with the lifestyle-related cancer. As a result,}

donations to the lifestyle related cancer become relatively unattractive as ↵pa+ (1 ↵) ¯pa

increases.

4.2.2.2 Donations to Cancer Treatment

Similarly to Section 4.2.1.2, the cancer rate for both cancer types is given bypc(d) =p0c (d).

cancer patients are also identical to the corresponding utilities in Section 4.2.1.2. The beliefs about pa and attitudes to ambiguity are the same as in the preceding section. Also similarly

to Section 4.2.2.1, the donor has ↵ maximin expected utility preferences.

Under these assumptions, we can write the non-choice egalitarian donor’s objective function as

Wn(d) = 2 (1 ps) [pc(d) +pc(100 d)].

while the choice egalitarian donor’s objective as

Wch(d) = 2 (1 ps)

⇥

pc(d) + 1 ↵pa+ (1 ↵) ¯pa pc(100 d)

⇤

.

The choice egalitarian donor’s optimal choice is given by

(1 ps)p0c

⇥ ₀

(d) 1 ↵pa+ (1 ↵) ¯pa 0(100 d)

⇤

= 0 (4.12)

Using the expression for Wn(d) and (4.12) we obtain:

Proposition 4 If the donor is non-choice egalitarian then her donation to the prevention of the lifestyle-related cancer will be una↵ected by her ambiguity aversion and her beliefs ⇥pa,p¯a

⇤

that cancer ls is lifestyle-related. Conversely, if the donor is choice egalitarian then

(a) a more ambiguity averse donor will donate more to the prevention of the lifestyle-related

cancer;

(b) an increase in the minimal likelihood of the lifestyle-related cancer, holding its degree of ambiguity constant, dpa=d¯pa>0will result in a decrease in the donation to the prevention

of the lifestyle-related cancer.

(c)an increase in the ambiguity of the lifestyle-related cancer, holding its minimal likelihood constant, d¯pa > 0 = dpa will result in a decrease in the donation to the prevention of the