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AMBIGUITY AND AMBIGUITY AND UNCERTAINTY IN ELLSBERG UNCERTAINTY IN ELLSBERG

AND SHACKLEAND SHACKLE

Marcello Basili and Carlo ZappiaMarcello Basili and Carlo Zappia

Department of EconomicsDepartment of EconomicsUniversity of SienaUniversity of Siena

FUR XIIFUR XIILuiss – Rome, 23-26 June, 2006Luiss – Rome, 23-26 June, 2006

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INTRODUCTIONINTRODUCTION

George L. S. Shackle and Daniel Ellsberg George L. S. Shackle and Daniel Ellsberg represent two main positions among the critics of represent two main positions among the critics of the forthcoming (at their time) mainstream in the forthcoming (at their time) mainstream in modern decision theory as represented by modern decision theory as represented by Savage’s Savage’s Foundations of Statistics Foundations of Statistics (1954)(1954)

Although both opposed maximisation of Although both opposed maximisation of (subjective) expected utility as a criterion for (subjective) expected utility as a criterion for choice under uncertainty, at first their theoretical choice under uncertainty, at first their theoretical enterprises appear to have few points in commonenterprises appear to have few points in common

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• Ellsberg (1961) introduced the notion of Ellsberg (1961) introduced the notion of “ambiguity” to refer to situations in which, due to “ambiguity” to refer to situations in which, due to lack of information, there is uncertainty about lack of information, there is uncertainty about probabilities on events. probabilities on events.

Ellsberg aimed to work in the footsteps of Knight Ellsberg aimed to work in the footsteps of Knight (1921) and his notion of “unmeasurable (1921) and his notion of “unmeasurable uncertainty,” but started from the analysis of uncertainty,” but started from the analysis of actual decisions in urn problemsactual decisions in urn problems

Though confounding and denoted by information Though confounding and denoted by information perceived as scanty, urn problems can be perceived as scanty, urn problems can be represented through an exhaustive list of the represented through an exhaustive list of the possible states of the world.possible states of the world.

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Shackle (1949 and 1955) insisted that the notion Shackle (1949 and 1955) insisted that the notion of uncertainty could not be reduced either to of uncertainty could not be reduced either to aleatory probability or subjective probabilityaleatory probability or subjective probability

Shackle rejected the use of probability measures Shackle rejected the use of probability measures in decision theory on the basis that the context of in decision theory on the basis that the context of “crucial” entrepreneurial decisions is “crucial” entrepreneurial decisions is characterized by the fact that the list of possible characterized by the fact that the list of possible states of the world known to the entrepreneur is states of the world known to the entrepreneur is not exhaustive not exhaustive

Keynesian authors have referred for long to Keynesian authors have referred for long to Shackle’s theory as the only formalised Shackle’s theory as the only formalised alternative to the mainstreamalternative to the mainstream

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This paper argues that Ellsberg’s and Shackle’s This paper argues that Ellsberg’s and Shackle’s frameworks for discussing the limits of the frameworks for discussing the limits of the (subjective) probabilistic approach to decision theory (subjective) probabilistic approach to decision theory are not as different as they may appear.are not as different as they may appear.

Both Ellsberg Both Ellsberg andand Shackle can be understood as main Shackle can be understood as main contributions of what today is called the non-additive contributions of what today is called the non-additive probability approachprobability approach to decisions under uncertainty to decisions under uncertainty

To stress the common elements in their theories To stress the common elements in their theories Keynes’s Keynes’s Treatise on Probability Treatise on Probability provides an essential provides an essential starting point (as recognised by Ellsberg in his 1962 starting point (as recognised by Ellsberg in his 1962 Ph.D. thesis, published only as late as 2001)Ph.D. thesis, published only as late as 2001)

Keynes’s rejection of well-defined probability Keynes’s rejection of well-defined probability functions and of maximisation as a guide to human functions and of maximisation as a guide to human conduct is shown to imply a reconsideration of what conduct is shown to imply a reconsideration of what probability theory can encompassprobability theory can encompass which inspired both which inspired both Ellsberg and ShackleEllsberg and Shackle

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KEYNES’S VIEWKEYNES’S VIEW

Keynes interpreted probability differently from Keynes interpreted probability differently from chance or frequency. Probability is a logical chance or frequency. Probability is a logical relation between two sets of propositionsrelation between two sets of propositions

The measurement of probabilities involves two The measurement of probabilities involves two magnitudes: the probability of an argument and magnitudes: the probability of an argument and the weight of the argumentthe weight of the argument

Measurement of probability means comparison of Measurement of probability means comparison of the arguments, for such a comparison is the arguments, for such a comparison is “theoretically possible, whether or not we are “theoretically possible, whether or not we are actually competent in every case to make”actually competent in every case to make”

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Keynes was well aware that the probabilities of Keynes was well aware that the probabilities of two quite different arguments can be two quite different arguments can be incomparable.incomparable.

Probabilities can be compared if they belong to Probabilities can be compared if they belong to the same series, that is, if they “belong to a the same series, that is, if they “belong to a single set of magnitude measurable in term of a single set of magnitude measurable in term of a common unit”common unit”

Probabilities are incomparable if they belong to Probabilities are incomparable if they belong to

two different arguments and one of them is not two different arguments and one of them is not (weakly) included in the other(weakly) included in the other

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Keynes’s idea about comparability is represented Keynes’s idea about comparability is represented by the diagram at page 42 of the by the diagram at page 42 of the TreatiseTreatise, , reproduced belowreproduced below

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On the horizontal axis there is the scale of On the horizontal axis there is the scale of probability, ranging from 0 to 1: every point on probability, ranging from 0 to 1: every point on this axes can be compared to other points, this axes can be compared to other points, because there exists a numerical representation because there exists a numerical representation of D-M degree of belief about a logical propositionof D-M degree of belief about a logical proposition

But in the plane depicted in the diagram there But in the plane depicted in the diagram there are also other different paths, starting from 0 and are also other different paths, starting from 0 and ending up in 1 but not lying on the straight line ending up in 1 but not lying on the straight line between the extremesbetween the extremes

Each point on every non-linear path identifies Each point on every non-linear path identifies what Keynes calls a what Keynes calls a non-numerical probabilitynon-numerical probability

From the point of view of modern theory of From the point of view of modern theory of decision, the Keynesian paths are nothing but decision, the Keynesian paths are nothing but distorted probabilities, that is, contraction or distorted probabilities, that is, contraction or expansion of prior linear probabilitiesexpansion of prior linear probabilities

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The second relevant aspect in the measurement The second relevant aspect in the measurement of an argument is in Keynes’s view, the weight of of an argument is in Keynes’s view, the weight of argumentargument

Keynes maintained that the weight of an Keynes maintained that the weight of an argument is correlated to its relevance, but is argument is correlated to its relevance, but is independent of its probabilityindependent of its probability. . For long, the For long, the prominent interpretation was to relate the weight prominent interpretation was to relate the weight of argument to the notion of second order of argument to the notion of second order probability distribution probability distribution

Keynes proposed a precise way to calculate it, Keynes proposed a precise way to calculate it, when he stated that the weight is:when he stated that the weight is:““the balance between the absolute amount of the the balance between the absolute amount of the relevant knowledge and of the relevant ignorance relevant knowledge and of the relevant ignorance respectively”respectively”

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In the language of modern decision theory (after In the language of modern decision theory (after Schmeidler and Dow and Werlang), given an event A, Schmeidler and Dow and Werlang), given an event A, the relevant ignorance can be defined asthe relevant ignorance can be defined as

that is, the difference between complete knowledge that is, the difference between complete knowledge and the probability of the occurrence of the event and the probability of the occurrence of the event plus the probability of its complement (negation of plus the probability of its complement (negation of the event)the event)

The weight of argument is then represented by The weight of argument is then represented by

that is, by the difference between what Keynes called that is, by the difference between what Keynes called the absolute amount of the relevant knowledge and the absolute amount of the relevant knowledge and the absolute amount of relevant ignorancethe absolute amount of relevant ignorance

))()((1 CAvAv

1

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If If vv is an is an additive measureadditive measure ( (probabilityprobability), the ), the relevant ignorance is zero and the weight of relevant ignorance is zero and the weight of argument is 1argument is 1. .

If If vv is a is a non-additive measurenon-additive measure ( (convex capacityconvex capacity), ), the relevant ignorance is different from zero and the relevant ignorance is different from zero and the weight of argument belongs to the interval the weight of argument belongs to the interval zero-onezero-one

This interpretation of Keynes’s thought entails that This interpretation of Keynes’s thought entails that the significance of the weight of argument emerges the significance of the weight of argument emerges only when the decision-maker is not endowed with only when the decision-maker is not endowed with a unique additive probability measure and does not a unique additive probability measure and does not behave as an expected utility maximizerbehave as an expected utility maximizer

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The third aspect of Keynes’s theory instrumental to The third aspect of Keynes’s theory instrumental to our argument emerges in Chapter 26 of the our argument emerges in Chapter 26 of the Treatise, Treatise, dealing with the application of probability to conductdealing with the application of probability to conduct

Keynes maintained that “mathematical expectation, Keynes maintained that “mathematical expectation, of goods or advantage are not always numerically of goods or advantage are not always numerically measurable, and hence that even if a meaning can be measurable, and hence that even if a meaning can be given to the sum of a series of non-numerical given to the sum of a series of non-numerical mathematical expectations, not every pair of such mathematical expectations, not every pair of such sums are numerically comparable in respect of more sums are numerically comparable in respect of more and less”and less”

As a result he (implicitly) rejected expected utility As a result he (implicitly) rejected expected utility maximisation from the outsetmaximisation from the outset

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ELLSBERG’S APPROACHELLSBERG’S APPROACH

• Ellsberg stressed that his findings imply that no Ellsberg stressed that his findings imply that no unique additive probability can account for the unique additive probability can account for the choices of unrepentant violators of the sure-thing choices of unrepentant violators of the sure-thing principleprinciple

• ““under most circumstances of decision-making under most circumstances of decision-making there may remain a sizeable subset Y° of there may remain a sizeable subset Y° of distributions … that still seem ‘reasonably distributions … that still seem ‘reasonably acceptable’ … that do not contradict your acceptable’ … that do not contradict your (‘vague’) opinions [and that] may yet be large, (‘vague’) opinions [and that] may yet be large, particularly when relevant information is particularly when relevant information is perceived as scanty, unreliable, contradictory, perceived as scanty, unreliable, contradictory, ambiguous” ambiguous”

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In 1961 Ellsberg had recommended as solution for In 1961 Ellsberg had recommended as solution for the paradox a weighted average of the expectation the paradox a weighted average of the expectation of the most reliable (“best guess”) probability of the most reliable (“best guess”) probability distribution and the maximin solutiondistribution and the maximin solution

Accordingly the decision rule adopted by Ellsberg Accordingly the decision rule adopted by Ellsberg was to associate with each act was to associate with each act xx the following index: the following index:

ρ E(ρ E(xx) + (1-ρ) min() + (1-ρ) min(x)x)

ρρ is a parameter that measures confidence and is a parameter that measures confidence and weighs the additive distribution that serves as a weighs the additive distribution that serves as a best estimate and all the other possible probability best estimate and all the other possible probability distributions assumed to be reasonable by the distributions assumed to be reasonable by the decision-maker under ambiguity (as in Hodges and decision-maker under ambiguity (as in Hodges and Lehmann 1952)Lehmann 1952)

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In the thesis of 1962 Ellsberg retained the idea of a In the thesis of 1962 Ellsberg retained the idea of a set of distributions over the states of the world, but set of distributions over the states of the world, but now applied to the restricted set of possible now applied to the restricted set of possible distributions the Hurwicz criterion. distributions the Hurwicz criterion.

Hurwicz (1951) had proposed to select the minimum Hurwicz (1951) had proposed to select the minimum and the maximum payoff to each given action and the maximum payoff to each given action xx, and , and then associates to each action the following index: then associates to each action the following index:

α max(x) + (1-α) min(x) α max(x) + (1-α) min(x) where the parameter alpha measures the where the parameter alpha measures the individual’s optimism (this is better know as Arrow-individual’s optimism (this is better know as Arrow-Hurwicz criterionHurwicz criterion

The new index Ellsberg proposed, called The new index Ellsberg proposed, called restricted restricted Bayes/Hurwicz criterionBayes/Hurwicz criterion, is:, is:

ρ E(ρ E(xx) + (1-ρ) [α max(x) + (1-α) min() + (1-ρ) [α max(x) + (1-α) min(xx)])]

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SHACKLE’S APPROACHSHACKLE’S APPROACH

• A decision-maker, typically an entrepreneur, has A decision-maker, typically an entrepreneur, has to choose among alternative “sequels” on the to choose among alternative “sequels” on the basis of two elements: the possible gains and basis of two elements: the possible gains and losses embedded in a sequel, called face-values, losses embedded in a sequel, called face-values, and a valuation of the “possibility” of the gains and a valuation of the “possibility” of the gains and losses, called potential surpriseand losses, called potential surprise

• The list of gains and losses is not complete so The list of gains and losses is not complete so (additive) probability theory cannot be applied(additive) probability theory cannot be applied

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Potential surprise can be considered as a degree Potential surprise can be considered as a degree of disbelief, or implausibility of the hypothesis that of disbelief, or implausibility of the hypothesis that supports the sequel. It can account for a “residual supports the sequel. It can account for a “residual hypothesis” (i.e. unanticipated event) hypothesis” (i.e. unanticipated event)

When the decision maker has to choose among When the decision maker has to choose among alternative sequels, she re-considers the face-alternative sequels, she re-considers the face-values of each sequel by their degree of potential values of each sequel by their degree of potential surprise.surprise.

Finally, Shackle defined a function Finally, Shackle defined a function σσ, called , called “degree of stimulus” (or “ascendancy function”) in “degree of stimulus” (or “ascendancy function”) in order to select two values, the focus-gain and the order to select two values, the focus-gain and the focus–loss, through which sequels are rankedfocus–loss, through which sequels are ranked

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After rejecting von Neumann-Morgenstern After rejecting von Neumann-Morgenstern expected utility maximization and considering expected utility maximization and considering Wald’s maximin too conservative, Shackle Wald’s maximin too conservative, Shackle presented his decision rule for the ranking of presented his decision rule for the ranking of sequels as follows:sequels as follows:

““there is surely a third [criterion] which is more there is surely a third [criterion] which is more plausible, and of more general analytical power, plausible, and of more general analytical power, than either of the two former, namely, that he [the than either of the two former, namely, that he [the decision-maker] will take into account both the decision-maker] will take into account both the ‘best possible’ and the ‘worst possible’ outcome of ‘best possible’ and the ‘worst possible’ outcome of each course of action and make these each course of action and make these pairspairs of of outcomes the basis of his decision”outcomes the basis of his decision”

So So Shackle’s criterion generalizes Hurwicz’sShackle’s criterion generalizes Hurwicz’s(better: Hurwicz is but a special case of Shackle(better: Hurwicz is but a special case of Shackle!)!)

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COMPARISON BETWEEN SHACKLE AND COMPARISON BETWEEN SHACKLE AND ELLSBERGELLSBERG

Shackle’s decision problem shares a crucial feature Shackle’s decision problem shares a crucial feature with the decision problem in which Ellsberg paradox with the decision problem in which Ellsberg paradox emerges. Both decision-making problems refer to an emerges. Both decision-making problems refer to an epistemic state that can be represented by the epistemic state that can be represented by the notion of ambiguity, which stands in the region notion of ambiguity, which stands in the region between the two extremes of complete ignorance between the two extremes of complete ignorance and riskand risk

At first the two problems seem quite different since, At first the two problems seem quite different since, in Ellsberg, the question is one of ambiguous in Ellsberg, the question is one of ambiguous probabilities, but with a complete list of all possible probabilities, but with a complete list of all possible events, while in Shackle, the question is one of events, while in Shackle, the question is one of providing an exhaustive list of possible events.providing an exhaustive list of possible events.

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But this difference concerns more the degree But this difference concerns more the degree rather than quality of uncertainty faced by the rather than quality of uncertainty faced by the decision-makerdecision-maker

Both Shackle and Ellsberg assume that the Both Shackle and Ellsberg assume that the decision-maker has partial knowledge about the decision-maker has partial knowledge about the future states of the world, and faces ambiguity in future states of the world, and faces ambiguity in the sense of our definition, that is, awareness of the sense of our definition, that is, awareness of the unreliability of a unique, additive probability the unreliability of a unique, additive probability distributiondistribution

What is relevant, as a result, is the representation What is relevant, as a result, is the representation of the of the small worldsmall world in which the decision-maker in which the decision-maker has act. From this viewpoint Shackle’s and the has act. From this viewpoint Shackle’s and the Ellsberg’s scenarios are analogous; both of them Ellsberg’s scenarios are analogous; both of them are miss-representation of the hypothetical are miss-representation of the hypothetical grand grand worldworld in which all future states of the world are in which all future states of the world are (potentially) completely described(potentially) completely described

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The grand world is the complete list of states The grand world is the complete list of states which are of concern to an individual. The small which are of concern to an individual. The small world is a construction derived from a certain world is a construction derived from a certain partition of the grand world into events, which partition of the grand world into events, which constitutes the states of the small-worldconstitutes the states of the small-world

A state in the small world is an event in the grand A state in the small world is an event in the grand world. A consequence in the small world is an act world. A consequence in the small world is an act in the grand worldin the grand world

Savage claimed that subjective expected theory Savage claimed that subjective expected theory should be applied only to small worlds. In fact, it is should be applied only to small worlds. In fact, it is only in small worlds that all possibilities can be only in small worlds that all possibilities can be exhaustively enumerated in advance, and all exhaustively enumerated in advance, and all implications of all possibilities explored in detailimplications of all possibilities explored in detail

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Savage added that:Savage added that:a small world is a a small world is a microcosm microcosm if if the probability of the probability of each state in the small world equals the each state in the small world equals the probability of corresponding event in the grand probability of corresponding event in the grand worldworld

But if small worlds are microcosm, the decision-But if small worlds are microcosm, the decision-maker is supposed to be able to enumerate maker is supposed to be able to enumerate exhaustively all possibilities in advance, and to exhaustively all possibilities in advance, and to explore all consequences in detail, though she explore all consequences in detail, though she works exclusively in a practical setting called the works exclusively in a practical setting called the small world.small world.

Thereby it is as if she had a sort of “divine” Thereby it is as if she had a sort of “divine” knowledgeknowledge of the outside worldof the outside world

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What happensWhat happens ifif the decision maker is not endowed the decision maker is not endowed with “divine knowledge” and the with “divine knowledge” and the set of states of the set of states of the small world that is relevant her is a miss-specified small world that is relevant her is a miss-specified representation because:representation because:

of possible missing states which are accounted for of possible missing states which are accounted for in the grand worldin the grand world (Shackle’s view)(Shackle’s view)

the decision-maker is unable to spit an event in the the decision-maker is unable to spit an event in the small word into its constitutive states in the grand small word into its constitutive states in the grand worldworld (Ellsberg’s view)(Ellsberg’s view)

??

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Shackle and Ellsberg talk of decision-making Shackle and Ellsberg talk of decision-making problems located in Savage’s small world. But of problems located in Savage’s small world. But of course their small worlds are not microcosmcourse their small worlds are not microcosm

To be precise, in both scenarios the decision-To be precise, in both scenarios the decision-maker is unable to enumerate the unique maker is unable to enumerate the unique mutually exclusively possible future states of the mutually exclusively possible future states of the world. Roughly speaking the decision-maker has world. Roughly speaking the decision-maker has only a rough representation (partition) of the set only a rough representation (partition) of the set of states of the world of states of the world

The non-additive probabilityThe non-additive probability approach provides approach provides the formal context for interpreting Ellsberg the formal context for interpreting Ellsberg andand ShackleShackle

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Depending on the epistemic condition of the Depending on the epistemic condition of the individual, the beliefs on the grand world may individual, the beliefs on the grand world may have a non-additive representation. In fact, if the have a non-additive representation. In fact, if the individual transfers a likelihood assigned to an individual transfers a likelihood assigned to an event in the small world to an event in the grand event in the small world to an event in the grand world, the implication is that she is unable to world, the implication is that she is unable to distribute beliefs across the elements of the distribute beliefs across the elements of the grand world.grand world.

The non-additivity of subjective probability The non-additivity of subjective probability measures becomes an expression of the limits of measures becomes an expression of the limits of the decision-maker’s understanding of the the decision-maker’s understanding of the possibilities of the world, as well as of her possibilities of the world, as well as of her awareness of these limits (Mukerji 1997).awareness of these limits (Mukerji 1997).

Hence it is legitimate to assume that an Hence it is legitimate to assume that an individual with a perception of the grand world as individual with a perception of the grand world as fuzzy, incomplete, or vague behaves as if she had fuzzy, incomplete, or vague behaves as if she had a non-additive prior rather than a well-defined a non-additive prior rather than a well-defined probabilityprobability

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But the analogy between Ellsberg’s and Shackle’s But the analogy between Ellsberg’s and Shackle’s problem and non-additive probability can be stressed problem and non-additive probability can be stressed as regards decision rules as well by making reference as regards decision rules as well by making reference to the modern developments of non-additive to the modern developments of non-additive probability theory as representing multiple priorsprobability theory as representing multiple priors

On the one hand, Gilboa and Schmeidler (1989) On the one hand, Gilboa and Schmeidler (1989) proposed to consider an individual that has opinions proposed to consider an individual that has opinions about the likelihood of different states, but she is not about the likelihood of different states, but she is not able to assign exact probabilities to them. According able to assign exact probabilities to them. According to their theory the decision maker has a convex set to their theory the decision maker has a convex set of subjective probability, which expresses the range of subjective probability, which expresses the range of probabilities she considers possible.of probabilities she considers possible.

Since the subjective probability is not unique there is Since the subjective probability is not unique there is a set of expected utilities for each action. Gilboa and a set of expected utilities for each action. Gilboa and Schmeidler then propose the following criterion: an Schmeidler then propose the following criterion: an action action aa is preferred to is preferred to bb if and only if the minimum if and only if the minimum possible value of the expected utility of possible value of the expected utility of aa is greater is greater than the minimum expected value of than the minimum expected value of bb..

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Following on Gardenfors and Sahlin (1982), they Following on Gardenfors and Sahlin (1982), they called this criterion maximin expected utility called this criterion maximin expected utility (MEU). If the set of probabilities consists only of a (MEU). If the set of probabilities consists only of a single probability distribution, maximin expected single probability distribution, maximin expected utility coincides with subjective expected utility. If utility coincides with subjective expected utility. If it consists of all possible probability distributions it consists of all possible probability distributions it coincides with Wald’s maximin.it coincides with Wald’s maximin.

On the other hand, Schmeidler showed that the On the other hand, Schmeidler showed that the

proper integral for a capacity is the Choquet proper integral for a capacity is the Choquet integral and proposed that decision-makers integral and proposed that decision-makers behave as if they maximise the Choquet integral behave as if they maximise the Choquet integral of their utility function.of their utility function.

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The Choquet integral with respect to a capacity The Choquet integral with respect to a capacity was known in the mathematical literature since was known in the mathematical literature since Choquet (1954) as a generalisation of the Choquet (1954) as a generalisation of the Lebesgue integral to a non-additive measure.Lebesgue integral to a non-additive measure.

Schmeidler applied the concept to decision under Schmeidler applied the concept to decision under uncertainty and used the Choquet integral of the uncertainty and used the Choquet integral of the non-additive measure as a generalisation of the non-additive measure as a generalisation of the mathematical expectation usually used in mathematical expectation usually used in expected utility models. This new procedure is expected utility models. This new procedure is usually referred to as Choquet expected utility usually referred to as Choquet expected utility (CEU).(CEU).

Furthermore Schmeidler provided an axiomatic Furthermore Schmeidler provided an axiomatic foundation for CEU by means of a representation foundation for CEU by means of a representation theorem that, in the same vein of Savage’s theorem that, in the same vein of Savage’s approach, made it possible to identify the non-approach, made it possible to identify the non-additive probability uniquely, and the utility additive probability uniquely, and the utility function up to a positive linear transformation.function up to a positive linear transformation.

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There is a close relationship between MEU and There is a close relationship between MEU and CEU that needs emphasising. CEU that needs emphasising.

Gilboa and Schmeidler (1994) showed that there Gilboa and Schmeidler (1994) showed that there is an isomorphism between a non-additive is an isomorphism between a non-additive probability measure and a convex set of additive probability measure and a convex set of additive probability measure. probability measure.

If the convex capacity that represents the If the convex capacity that represents the decision-maker’s beliefs has a non-empty core of decision-maker’s beliefs has a non-empty core of additive measures, than the CEU with respect to additive measures, than the CEU with respect to the capacity equals the maximin expected utility the capacity equals the maximin expected utility of the set of additive measures, that is, the of the set of additive measures, that is, the subjective expected utility with respect to the less subjective expected utility with respect to the less favourable probability distribution in the core.favourable probability distribution in the core.

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MEU suggests that the non-additive probability is MEU suggests that the non-additive probability is the lower bound of what the “real” additive the lower bound of what the “real” additive probability might be, and concentrates on the worst probability might be, and concentrates on the worst cases, regardless of any consideration relative to cases, regardless of any consideration relative to the individual’s degree of confidence in her the individual’s degree of confidence in her probability assessment.probability assessment.

This limitation is overcome by a generalized version This limitation is overcome by a generalized version of MEU called of MEU called -maxmin expected utility (-maxmin expected utility (-MEU). In -MEU). In this theory a crucial role is played by the parameter this theory a crucial role is played by the parameter [0,1], which expresses, exactly as in Arrow-[0,1], which expresses, exactly as in Arrow-Hurwicz, the decision maker’s ambiguity attitude.Hurwicz, the decision maker’s ambiguity attitude.

The The -MEU emphasises the decision maker’s degree -MEU emphasises the decision maker’s degree of ambiguity perception, and involves a valuation of of ambiguity perception, and involves a valuation of the quality of the decision maker’s information.the quality of the decision maker’s information.

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CONCLUDING REMARKSCONCLUDING REMARKS

Shackle and Ellsberg talk of decision-making Shackle and Ellsberg talk of decision-making problems located in Savage’s small world. Their problems located in Savage’s small world. Their small worlds are such that, in both cases, the small worlds are such that, in both cases, the decision-maker is unable to enumerate the unique decision-maker is unable to enumerate the unique mutually exclusively possible future states of the mutually exclusively possible future states of the worldworld

Shackle and Ellsberg were substantially referring Shackle and Ellsberg were substantially referring to a model of ambiguity aversion currently known to a model of ambiguity aversion currently known as the maxmin expected utility model (MEU) and as the maxmin expected utility model (MEU) and Choquet expectedChoquet expected

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Recently a slightly different decision rule, the Recently a slightly different decision rule, the α-α-MEU approachMEU approach, has been proposed as an outcome , has been proposed as an outcome of these developmentsof these developments

Ghirardato, Maccheroni and Marinacci (2004) Ghirardato, Maccheroni and Marinacci (2004) assume that the decision-maker’s ambiguity is assume that the decision-maker’s ambiguity is expressed by a set of additive probability expressed by a set of additive probability distributions (multiple priors) and the parameter α distributions (multiple priors) and the parameter α represents her ambiguity attitude, that isrepresents her ambiguity attitude, that is

)(min)1()(max)( fuEfuEfI

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The α-MEU preference model is a generalization of The α-MEU preference model is a generalization of the Hurwicz’s maxmin functional in which the the Hurwicz’s maxmin functional in which the parameter α is constantparameter α is constant

Although Shackle did not explicitly set the Although Shackle did not explicitly set the functional of the criterion he was proposing, it is functional of the criterion he was proposing, it is straightforward to note that it overlaps with the straightforward to note that it overlaps with the --MEU functional. MEU functional.

As a result both Shackle’s and Ellsberg’s approach As a result both Shackle’s and Ellsberg’s approach can be considered as variations of Hurwicz’s can be considered as variations of Hurwicz’s maxmin expected utility criterion (α-MEU)maxmin expected utility criterion (α-MEU)

On these grounds the ambiguity surrounding the On these grounds the ambiguity surrounding the decision maker in Ellsberg’s urn experiment can be decision maker in Ellsberg’s urn experiment can be deemed analogous to the uncertainty faced by deemed analogous to the uncertainty faced by Shackle’s entrepreneur taking unique decisionsShackle’s entrepreneur taking unique decisions