A Note on Choice under Ambiguity with Optimism on Windfall Gains and Pessimism on Catastrophic Losses

1
A Note on Choice under 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

2

• The paper investigates on decision-making process
  involving both risk and ambiguity
• Attitude towards ambiguity generally (i.e. in
  CPT) pessimism on gains, optimism on losses

3

• 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?

4

• 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

5
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
  Inter-Continental-Ballistic-Missiles (ICBMs)
  owned by the Red Soviet Army

6

• 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

7

• Mr. the President and his Staff assumed that URSS
  had thousands of ICBM and started the production
  of one thousand solid-fueled Minuteman missiles

8

• 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)

9

• 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

10

• We believe that all these kinds of behavior can
  be represented by a Choquet Integral (CI) that is
  sufficiently general to represent decision
  makers optimism towards unexpected gains and
  pessimism with respect to unusual losses

11

• 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-makers beliefs assume a simple yet
  intuitive structure, namely symmetry towards risk
  and ambiguity, and faces situations that are
  fully ambiguous

12

• 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 decision-making under
  ambiguity

13

• The RBHC is a generalization of Hurwiczs
  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)

14
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

15
(No Transcript)
16
(No Transcript)
17
ASSUMPTIONS
18
(No Transcript)
19
(No Transcript)
20
(No Transcript)
21
(No Transcript)
22
(No Transcript)
23
(No Transcript)
24
(No Transcript)
25

• Definition 3 means that the decision-maker 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 decision-maker faces
  the same level of ambiguity in the unfamiliar
  world

26
(No Transcript)
27
(No Transcript)
28

• 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 ?, 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

29
CONCLUDING REMARKS

• Since capacities v-, p, 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.

30

• 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 ?, that represents
  the degree of confidence the decision-maker
  maintains on the probabilistic judgment (Dow and
  Werlang 1994, Marinacci 2000)

31

• In many decision problems simple (and dual
  simple) capacities provide a intuitive and easily
  tractable framework that can be sufficient to
  express decision-makers 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-makers behavior (e.g. Ellsbergs
  two-color urn paradox (Ellsberg 1961)
• Our representation in the corollary mimics
  Ellsbergs RBHC of choice under ambiguity

32

• Finally, our approach might induce several useful
  implementations
• 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) - a-MEU approach - but it is a
  combination between the extreme outcomes and
  mathematical expectation of all the possible
  results attached to each act
• In behavioral finance to extend application of
  prospect theory approach to explain investors
  behavior in financial markets