A Note on Choice under A Note on Choice under Ambiguity with Optimism on Ambiguity with Optimism on

Windfall Gains and Pessimism Windfall Gains and Pessimism on Catastrophic Losseson Catastrophic Losses

Marcello BasiliMarcello BasiliDepartment of Economics, University of SienaDepartment of Economics, University of Siena

Alain ChateauneufAlain ChateauneufCERMSEM, University of Paris-ICERMSEM, University of Paris-I

Fulvio FontiniFulvio FontiniDepartment of Economics, University of PaduaDepartment of Economics, University of Padua

FUR XIIFUR XII 22-26 June 22-26 June 2006 at 2006 at LUISS in ROMELUISS in ROME

The paper investigates on decision-The paper investigates on decision-making process involving both risk and making process involving both risk and ambiguityambiguity

Attitude towards ambiguity: generally (i.e. Attitude towards ambiguity: generally (i.e. in CPT) pessimism on gains, optimism on in CPT) pessimism on gains, optimism on losseslosses

Main Question:Main Question:

Is it plausible to conceive the opposite, Is it plausible to conceive the opposite, that is that is pessimism on extreme losses pessimism on extreme losses and optimism on windfall gainsand optimism on windfall gains??

Second QuestionSecond Question

In the case of an affirmative answer are In the case of an affirmative answer are there relevant consequences?there relevant consequences?

There are at least three main sources There are at least three main sources that support our question:that support our question:

a) a) evidenceevidence (Etchart-Vincent 2004 (Etchart-Vincent 2004 JRU, Levy and Levy 2002 JRU, Levy and Levy 2002

Man.Sc.); Man.Sc.);

b) b) introspectionintrospection;;

c) c) anecdotal speechesanecdotal speeches

ANECDOTAL SPEECHANECDOTAL SPEECH

Ellsberg refereed the situation in which Mr. the Ellsberg refereed the situation in which Mr. the President of USA had to decide about the President of USA had to decide about the development of nuclear weapons to face the development of nuclear weapons to face the menace of URSS in 60's menace of URSS in 60's

Ellsberg referred of some meetings in which Ellsberg referred of some meetings in which there were all the US Secrete Services (CIA, there were all the US Secrete Services (CIA, FBI, US-Navy, US-Army, USAF etc.) and Mr. FBI, US-Navy, US-Army, USAF etc.) and Mr. The President of USA and his Staff asked them The President of USA and his Staff asked them a reliable estimation of the number of Inter-a reliable estimation of the number of Inter-Continental-Ballistic-Missiles (ICBMs) owned by Continental-Ballistic-Missiles (ICBMs) owned by the Red Soviet Armythe Red Soviet Army

The answers were different and went The answers were different and went from thousands (USAF) to a handful (US-from thousands (USAF) to a handful (US-Navy)Navy)

In a such situation, characterized by a In a such situation, characterized by a set of probability distributions, none of set of probability distributions, none of which fully reliable, about possible states which fully reliable, about possible states of the world, Mr. the President and his of the world, Mr. the President and his Staff Staff were pessimisticwere pessimistic and based their and based their decision of developing the ICBM rum on decision of developing the ICBM rum on the the worst possible scenarioworst possible scenario

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

In the fall of 1961, as Ellsberg reported, In the fall of 1961, as Ellsberg reported, a revised highly secret report set that a revised highly secret report set that ""the missile gap favoring the Soviets the missile gap favoring the Soviets had been a fantasy. There was a gap, had been a fantasy. There was a gap, but it was currently ten to one in our but it was currently ten to one in our favor. Our 40 Atlas and Titan ICBMs favor. Our 40 Atlas and Titan ICBMs were matched by 4 Soviet SS-6 ICBMs were matched by 4 Soviet SS-6 ICBMs at one launching site at Plesetskat one launching site at Plesetsk" " (Ellsberg 2002, p. 32)(Ellsberg 2002, p. 32)

On the basis of this and other real reports On the basis of this and other real reports we think that is possible to assume that:we think that is possible to assume that:

differently from the most part of differently from the most part of experimental evidence (in which losses experimental evidence (in which losses are generally underestimated), people are generally underestimated), people has a pessimistic attitude when face has a pessimistic attitude when face catastrophic lossescatastrophic losses

Symmetrically, it seems meaningful for us Symmetrically, it seems meaningful for us to suppose that persons have an to suppose that persons have an optimistic attitude with respect to the optimistic attitude with respect to the windfall gainswindfall gains

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

Moreover, by restricting attention to a specific Moreover, by restricting attention to a specific attitude towards uncertainty (i.e. a specific attitude towards uncertainty (i.e. a specific sub-set of capacities) we show that our CI sub-set of capacities) we show that our CI assumes an intuitive representation and can assumes an intuitive representation and can be further simplified into a linear combination be further simplified into a linear combination of the expected utility and the utility of the of the expected utility and the utility of the most extreme outcomes, the highest windfall most extreme outcomes, the highest windfall gain and the worst catastrophic loss, gain and the worst catastrophic loss, whenever the decision-maker’s beliefs whenever the decision-maker’s beliefs assume a simple yet intuitive structure, assume a simple yet intuitive structure, namely symmetry towards risk and ambiguity, namely symmetry towards risk and ambiguity, and faces situations that are fully ambiguousand faces situations that are fully ambiguous

Our approach has some similarity with Our approach has some similarity with the Restricted Bayes-Hurwicz Criterion the Restricted Bayes-Hurwicz Criterion (RBHC) proposed by Ellsberg in his (RBHC) proposed by Ellsberg in his Ph.D. Dissertation (Ellsberg, 2001)Ph.D. Dissertation (Ellsberg, 2001)

It is the most general criterion of choice It is the most general criterion of choice recommended by Ellsberg in decision-recommended by Ellsberg in decision-making under ambiguity making under ambiguity

The RBHC is a generalization of Hurwicz’s The RBHC is a generalization of Hurwicz’s Criterion or the Maximin Criterion when the Criterion or the Maximin Criterion when the decision-maker not only considers "the decision-maker not only considers "the reliability, credibility or adequacy of reliability, credibility or adequacy of information, experience, advice, intuition information, experience, advice, intuition taken as a whole: not about the relative taken as a whole: not about the relative support it may give to one hypothesis as support it may give to one hypothesis as opposed to another, but about its ability to opposed to another, but about its ability to lend support to any hypothesis - any set of lend support to any hypothesis - any set of definite options - at all" (Ellsberg 2001, definite options - at all" (Ellsberg 2001, p.192), but also "relative willingness to rely p.192), but also "relative willingness to rely upon it in [her] decision-making; and various upon it in [her] decision-making; and various factors enter [her] decision criterion in linear factors enter [her] decision criterion in linear combination" (Ellsberg 2001, p. 193)combination" (Ellsberg 2001, p. 193)

SET-UPSET-UP

The decision-maker has well-defined risk and The decision-maker has well-defined risk and ambiguity attitudeambiguity attitude

The capacity is strictly non-additive on unfamiliar The capacity is strictly non-additive on unfamiliar events, because of ambiguity attitude, and events, because of ambiguity attitude, and additive on events related to customary additive on events related to customary outcomes outcomes

As a result, the decision-maker perceives As a result, the decision-maker perceives genuine ambiguity with respect to unfamiliar genuine ambiguity with respect to unfamiliar losses and gains and is ambiguity neutral across losses and gains and is ambiguity neutral across the customary outcomesthe customary outcomes

ASSUMPTIONSASSUMPTIONS

Definition 3 means that the decision-Definition 3 means that the decision-maker takes m and M as being equally maker takes m and M as being equally bad and good, in the sense that she bad and good, in the sense that she takes the biggest familiar loss and the takes the biggest familiar loss and the highest familiar gain as being equally highest familiar gain as being equally distant from zerodistant from zero

Definition 4 means that the decision-Definition 4 means that the decision-maker faces the same level of ambiguity maker faces the same level of ambiguity in the unfamiliar worldin the unfamiliar world

Corollary 3Corollary 3 shows that the decision-maker shows that the decision-maker represents her beliefs according to a represents her beliefs according to a functional which is a linear combination of the functional which is a linear combination of the expected outcome over all gains and losses expected outcome over all gains and losses and the best/worst ones, where the latter and the best/worst ones, where the latter encompass the whole weight of ambiguityencompass the whole weight of ambiguity

The right hand side of the CI in (9) shows that The right hand side of the CI in (9) shows that the decision-maker balances the best windfall the decision-maker balances the best windfall gain and the worst catastrophic loss that she gain and the worst catastrophic loss that she is going to bearis going to bear

For a given degree of confidence γ, she is For a given degree of confidence γ, she is more willing to undertake an act that might more willing to undertake an act that might lead to truly unusual consequences if the lead to truly unusual consequences if the former is bigger than the latter, and vice former is bigger than the latter, and vice versaversa

CONCLUDING REMARKSCONCLUDING REMARKS

Since capacities v−, π, v+ are not Since capacities v−, π, v+ are not restricted, Theorem 1 is general and it restricted, Theorem 1 is general and it generalizes the usual CI (e.g., generalizes the usual CI (e.g., Schmeidler 1989) by allowing Schmeidler 1989) by allowing partitioning the set of outcomes into partitioning the set of outcomes into familiar and unfamiliar ones and taking familiar and unfamiliar ones and taking into account both gains and losses. into account both gains and losses.

The generality of the result of Theorem The generality of the result of Theorem 1 is reduced in Theorem 2, where v+, 1 is reduced in Theorem 2, where v+, v− are restricted to be simple and v− are restricted to be simple and simple dual capacities, respectively, simple dual capacities, respectively, parametrized by γ, that represents the parametrized by γ, that represents the degree of confidence the decision-degree of confidence the decision-maker maintains on the probabilistic maker maintains on the probabilistic judgment (Dow and Werlang 1994, judgment (Dow and Werlang 1994, Marinacci 2000)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 two-color 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 xtreme events, that is disasters and catastrophes (windfall gains, seldom) that are disasters and catastrophes (windfall gains, seldom) that are characterized by very small or ambiguous probabilities of characterized by very small or ambiguous probabilities of occurring. occurring. The new notion of the precautionary principle based on The new notion of the precautionary principle based on our approach is not a simple convex combination between maximin our approach is not a simple convex combination between maximin ((conservative actconservative act) and maximax criterion () and maximax criterion (dissipative actdissipative act) - ) - α-MEU α-MEU approachapproach - but it is a combination between the extreme outcomes - but it is a combination between the extreme outcomes and mathematical expectation of all the possible results attached to and mathematical expectation of all the possible results attached to each act;each act;

2.2. In behavioral finance to extend application of prospect theory In behavioral finance to extend application of prospect theory approach to explain investors behavior in financial marketsapproach to explain investors behavior in financial markets