Ambiguity with Optimism on
Windfall Gains and Pessimism
on Catastrophic Losses
Marcello Basili
Department of Economics, University of Siena
Alain Chateauneuf
CERMSEM, University of Paris-I
Fulvio Fontini
Department of Economics, University of Padua
FUR XII 22-26 June 2006 at LUISS in ROME
 The paper investigates on decisionmaking process involving both risk and
ambiguity
 Attitude towards ambiguity: generally (i.e.
in CPT) pessimism on gains, optimism on
losses
 Main Question:
Is it plausible to conceive the opposite,
that is pessimism on extreme losses
and optimism on windfall gains?
 Second Question
In the case of an affirmative answer are
there relevant consequences?
There are at least three main sources
that support our question:
a) evidence (Etchart-Vincent 2004
JRU, Levy and Levy 2002 Man.Sc.);
b) introspection;
c) anecdotal speeches
ANECDOTAL SPEECH
 Ellsberg refereed the situation in which Mr. the
President of USA had to decide about the
development of nuclear weapons to face the
menace of URSS in 60's
 Ellsberg referred of some meetings in which
there were all the US Secrete Services (CIA,
FBI, US-Navy, US-Army, USAF etc.) and Mr. The
President of USA and his Staff asked them a
reliable estimation of the number of InterContinental-Ballistic-Missiles (ICBMs) owned by
the Red Soviet Army
 The answers were different and went
from thousands (USAF) to a handful
(US-Navy)
 In a such situation, characterized by a
set of probability distributions, none of
which fully reliable, about possible
states of the world, Mr. the President
and his Staff were pessimistic and
based their decision of developing the
ICBM rum on the worst possible
scenario
 Mr. the President and his Staff
assumed that URSS had thousands
of ICBM and started the production
of one thousand solid-fueled
Minuteman missiles
 In the fall of 1961, as Ellsberg reported,
a revised highly secret report set that
"the missile gap favoring the Soviets
had been a fantasy. There was a gap,
but it was currently ten to one in our
favor. Our 40 Atlas and Titan ICBMs
were matched by 4 Soviet SS-6 ICBMs
at one launching site at Plesetsk"
(Ellsberg 2002, p. 32)
 On the basis of this and other real
reports we think that is possible to
assume that:
 differently from the most part of
experimental evidence (in which losses
are generally underestimated), people
has a pessimistic attitude when face
catastrophic losses
 Symmetrically, it seems meaningful for
us to suppose that persons have an
optimistic attitude with respect to the
windfall gains
 We believe that all these kinds of
behavior can be represented by a
Choquet Integral (CI) that is sufficiently
general to represent decision maker’s
optimism towards unexpected gains and
pessimism with respect to unusual
losses
 Moreover, by restricting attention to a specific
attitude towards uncertainty (i.e. a specific
sub-set of capacities) we show that our CI
assumes an intuitive representation and can
be further simplified into a linear combination
of the expected utility and the utility of the
most extreme outcomes, the highest windfall
gain and the worst catastrophic loss,
whenever the decision-maker’s beliefs
assume a simple yet intuitive structure,
namely symmetry towards risk and ambiguity,
and faces situations that are fully ambiguous
 Our approach has some similarity with
the Restricted Bayes-Hurwicz Criterion
(RBHC) proposed by Ellsberg in his
Ph.D. Dissertation (Ellsberg, 2001)
 It is the most general criterion of choice
recommended by Ellsberg in decisionmaking under ambiguity
 The RBHC is a generalization of Hurwicz’s
Criterion or the Maximin Criterion when the
decision-maker not only considers "the
reliability, credibility or adequacy of
information, experience, advice, intuition
taken as a whole: not about the relative
support it may give to one hypothesis as
opposed to another, but about its ability to
lend support to any hypothesis - any set of
definite options - at all" (Ellsberg 2001,
p.192), but also "relative willingness to rely
upon it in [her] decision-making; and various
factors enter [her] decision criterion in linear
combination" (Ellsberg 2001, p. 193)
SET-UP
 The decision-maker has well-defined risk and
ambiguity attitude
 The capacity is strictly non-additive on unfamiliar
events, because of ambiguity attitude, and
additive on events related to customary
outcomes
 As a result, the decision-maker perceives
genuine ambiguity with respect to unfamiliar
losses and gains and is ambiguity neutral across
the customary outcomes
ASSUMPTIONS
 Definition 3 means that the decisionmaker takes m and M as being equally
bad and good, in the sense that she
takes the biggest familiar loss and the
highest familiar gain as being equally
distant from zero
 Definition 4 means that the decisionmaker faces the same level of ambiguity
in the unfamiliar world
 Corollary 3 shows that the decision-maker
represents her beliefs according to a
functional which is a linear combination of the
expected outcome over all gains and losses
and the best/worst ones, where the latter
encompass the whole weight of ambiguity
 The right hand side of the CI in (9) shows that
the decision-maker balances the best windfall
gain and the worst catastrophic loss that she
is going to bear
 For a given degree of confidence γ, she is
more willing to undertake an act that might
lead to truly unusual consequences if the
former is bigger than the latter, and vice versa
CONCLUDING REMARKS
 Since capacities v−, π, v+ are not
restricted, Theorem 1 is general and it
generalizes the usual CI (e.g.,
Schmeidler 1989) by allowing
partitioning the set of outcomes into
familiar and unfamiliar ones and taking
into account both gains and losses.
 The generality of the result of Theorem
1 is reduced in Theorem 2, where v+,
v− are restricted to be simple and
simple dual capacities, respectively,
parametrized by γ, that represents the
degree of confidence the decisionmaker maintains on the probabilistic
judgment (Dow and Werlang 1994,
Marinacci 2000)
 In many decision problems simple (and dual
simple) capacities provide a intuitive and easily
tractable framework that can be sufficient to
express decision-maker’s attitude towards
ambiguity, whenever one can clearly distinguish
between ambiguity and risk attitude and identify
the role that both subjective evaluation of
outcomes and beliefs play in assessing the
decision-maker’s behavior (e.g. Ellsberg’s twocolor urn paradox (Ellsberg 1961)
 Our representation in the corollary mimics
Ellsberg’s RBHC of choice under ambiguity
 Finally, our approach might induce several useful implementations:
1. in situations that require the application of the precautionary
principle, when the decision-maker faces extreme events, that is
disasters and catastrophes (windfall gains, seldom) that are
characterized by very small or ambiguous probabilities of occurring.
The new notion of the precautionary principle based on our
approach is not a simple convex combination between maximin
(conservative act) and maximax criterion (dissipative act) - α-MEU
approach - but it is a combination between the extreme outcomes
and mathematical expectation of all the possible results attached to
each act;
2. In behavioral finance to extend application of prospect theory
approach to explain investors behavior in financial markets