Marcello Basili Department of Economics, University of Siena Alain Chateauneuf

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 XII22-26 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 maker’s 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-maker’s 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 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)

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

17. ASSUMPTIONS

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

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 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

29. 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.

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 γ, 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-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 two-color urn paradox (Ellsberg 1961) • Our representation in the corollary mimics Ellsberg’s 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) - α-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