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Manuscript Number: Title: Regret theory with uncertain subjective probabilities and pessimism Article Type: Original Research Section/Category: Keywords: Ambiguity; Ellsberg's paradox; Event-Splitting effects; Intuitive probabilities; Pessimism; Regret Theory. Corresponding Author: Mr. Jérôme Villion, Corresponding Author's Institution: CREM First Author: Jérôme Villion Order of Authors: Jérôme Villion Manuscript Region of Origin: Abstract: This paper presents a generalization of Regret theory when ambiguity matters. We use an intuitive approach of belief and we characterize decision-maker's attitude in ambiguity through notions of optimism and pessimism. The model does not fulfil the regret-dominance but it respects, under some conditions, the simple dominance and a property called regret-dominance-under-uncertainty. The model violates the surething principle and provides an explanation of the Ellsberg paradox when the decision-maker is pessimistic. A property called regret-reduction is also violated which makes possible an explanation of event-splitting effects; but, contrary to Starmer and Sugden (1993)'s stripped-down prospect theory and to Humphrey (1998)'s models, our model respects a property called identity-regrouping principle. We present some experimental results and we deal with methodological and normative questions.

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JÉRÔME VILLION

REGRET THEORY WITH UNCERTAIN SUBJECTIVE PROBABILITIES AND PESSIMISM

ABSTRACT. This paper presents a generalization of Regret theory when ambiguity matters. We use an intuitive approach of belief and we characterize decisionmaker’s attitude in ambiguity through notions of optimism and pessimism. The model does not fulfil the regret-dominance but it respects, under some conditions, the simple dominance and a property called regret-dominance-under-uncertainty. The model violates the sure-thing principle and provides an explanation of the Ellsberg paradox when the decision-maker is pessimistic. A property called regretreduction is also violated which makes possible an explanation of event-splitting effects; but, contrary to Starmer and Sugden (1993)’s stripped-down prospect theory and to Humphrey (1998)’s models, our model respects a property called identity-regrouping principle. We present some experimental results and we deal with methodological and normative questions.

KEY WORDS: Ambiguity; Ellsberg’s paradox; Event-Splitting effects; Intuitive probabilities; Pessimism; Regret Theory. 1. INTRODUCTION Regret theory (Loomes and Sugden (1982, 1987)), denoted RTc hereafter, has a special place in decision theory under uncertainty. First, while most of the models that generalize expected utility weaken the independence axiom, RTc violates transitivity. Moreover, RTc does not satisfy either reduction (or equivalence) nor first order stochastic dominance (or monotonicity). From a normative point of view, first 1

order stochastic dominance, reduction and, specially, transitivity are generally considered as properties that preferences should satisfy. Nevertheless, Loomes and Sugden maintain that RTc represents preferences that are consistent. The argument was at first based on the fact that regret is a psychological state that is not irrational to take into account in decision (Loomes and Sugden (1982), Sugden (1985, 1991)). Later, Sugden (1993) has provided an axiomatic foundation for RTc. Second, the psychological experience of regret creates a within-event / between-acts effect. Then, RTc does not assign values independently to individual acts. This is a reason why RTc is a “nonconventional theory” (Starmer (2000)). Third, RTc is one of the few models that can explain the preference reversal phenomenon (Lichtenstein and Slovic (1971), Lindman (1971); see Loomes and Sugden (1983), Loomes and alii. (1989, 1991), Loomes and Taylor (1992) and Starmer and Sugden (1998)). At the same time, RTc provides an explanation of the Allais paradox (Allais (1953)). Actually, RTc is consistent with the common ratio effect (see Loomes and Sugden (1987b) and Loomes (1988)) and, for statistically independent prospects, with the common consequence effect (see Starmer (1992)). However, because RTc satisfies the sure-thing principle (Savage (1954)), the model cannot explain a well-known phenomenon in decision under uncertainty. This is the so-called Ellsberg paradox (Ellsberg (1961)). Our main aim is to generalize RTc so as to provide an explanation of the Ellsberg paradox. The model we propose, denoted RTu, is based on: an intuitive approach of beliefs, a representation of beliefs that is not a probability measure and a psychological hypothesis of pessimism. The intuitive approach of beliefs is nonconventional in economics. In a fully subjective view, this approach does not allow the elaboration of a decision model uniquely from axioms on the binary relation on acts. Most of the models follow on the contrary the route of Ramsey (1926) and Savage (1954) and retain a decision2

oriented approach in which the belief is defined from the preferences on acts. The intuitive approach has been however widely developed in mathematical, statistical and philosophical work (see Fishburn (1986a)). It provides a measure of belief as a likelihood of events, independently of decisions. It contrasts with decision-oriented approach for which this likelihood is unattainable: only a propensity to bet can be reached; that is the belief cannot be separated from all the characteristics of the decision problem (see Villion (2004b) for a discussion). Representation of beliefs through measures more general than (addititive) probability is quite usual in the literature. But, we can note again a difference between decision-oriented and intuitive approaches. In the first ones, the measure of belief alternative to probability is usually a capacity (e.g. Schmeidler (1989), Ghirardato and Marinacci (2001)) or a set of probability measures (Gilboa and Schmeidler (1989)). In the second ones, lower and upper probability measures, eventually coupled with a probability measure, are frequently adopted for belief representation

(e.g.

Koopman (1940),

Good (1962),

Suppes (1974)

and

Fishburn (1986b)). We also retain this latter type of representation for the model we propose. The pessimism hypothesis we introduce can be informally defined in the following way. A decision maker is pessimistic if, for all acts, she assigns to every event leading to relatively favorable (unfavorable) consequences a decision weight that is smaller (greater) than its subjective probability, between the limits given by the lower and upper probabilities of the event. Because what we call a favorable (unfavorable) consequence is one that yields a utility greater (smaller) than the expected utility of the act, pessimism creates an intra-act / between-event dependence. Then, RTu violates the sure-thing principle and can explain the Ellsberg paradox. In other respects, our intuitive approach of beliefs leads to a notion of pessimism that is different from the ones we find generally in the 3

literature on decision under uncertainty. In most of the models, pessimism or ambiguity

aversion

characteristics

of

or

the

uncertainty preference

aversion

relation

on

are

defined

acts

(e.g.

through

some

Schmeildler (1989),

Epstein (1999), Ghirardato and Marinacci (2002); see Cohen and Tallon (2000) for a review) and generally find expression in the form of the belief representation. In RTu, there is a strict separation between the beliefs and the behavior of the decision-maker faced with uncertainty, this behavior being represented through the form of the decision weight. The rest of the paper is organized as follow. Section 2 presents the RTu model. Section 3 states the properties respected by RTu. In section 4, we show that RTu provides an explanation of the Ellsberg paradox and event-splitting effects. Some results of an experimentation we have conducted are presented in section 5. Section 6 concludes with a discussion on methodological implications of RTu. 2. THE MODEL

2.1. Notations and definitions S is the finite set of states of the world s,s1,...,sj,... Subsets of S are called events, denoted A,B,C,D,B1,B2,... 2 S = { A: A ⊆ S } is the set of all events. ∅ is the empty event. S is the universal event. A (simple) probability measure π is a realvalued

set-function

on

2S

such

that

π(A) ≥0,

∀A∈ 2S ,

π(S) =1

and

π(A∪B) = π(A) + π(B) for all A,B∈ 2S such that A∩B = ∅. X is the set of consequences x,y,z,u,v,w,x1,x2,...,xi,... X M ⊆ X is the set of monetary consequences. F is the set of simple acts f,g,h,f1,f2,... A simple act is a function f from S to X with

{ f ( s ): s ∈ S}

finite. A constant act is an act f such that f(s)=x, ∀s∈S with x∈X, and we denote f=x. A constant act on A is an act f such that f(s)=x, ∀s∈A, with x∈X, and we denote 4

fA=x. For all finite set of acts H⊆F, a finite partition of S is a constant partition for H if for all f in H and all event A in the partition, f is constant on A and we denote fA the consequence of f on A. fA=gA means f(s)=g(s), ∀s∈A. And, xAy denotes the act which yields x on A and y on Ac, that is the act f such that fA=x and f AC = y . We model the decision-maker’s preferences on F by a binary relation and ~ respectively denote the asymmetric and symmetric parts of

f. ~

As usual,

f

f. ~

2.2. Intuitive beliefs We suppose that there exists, for each individual, a probability measure π, and two other real-valued set-functions on 2S called a lower probability measure, denoted π * , and an upper probability measure, denoted π ∗ . For each event A, π(A),

π ∗ (A) and π ∗ (A) form what we call uncertain subjective probabilities and measure the belief of the individual about A. This is an intuitive belief, independent of any decision problem. We impose the following conditions. B1: π ∗ (∅) = π ∗ (∅) = 0 and π ∗ (S) = π ∗ (S) = 1. B2: ∀A∈ 2 S , π ∗ (A) = 1 - π ∗ (Ac ). B3: ∀A,B∈ 2 S , (A∩B = ∅) ⇒ π ∗ (A∪B) ≥ π ∗ (A) + π ∗ (B) (super-additivity). B4: ∀A,B∈ 2 S , (A∩B = ∅) ⇒ π ∗ (A∪B) ≤ π ∗ (A) + π ∗ (B) (sub-additivity). B5: ∀A∈ 2 S , π ∗ (A) ≤ π(A) ≤ π ∗ (A). Conditions B1 to B5 can be seen either, in Good (1962)’s tradition, as quantitative properties that seem natural for a model of lower and upper

5

probabilities, or, in Koopman (1940)’s tradition, as properties inferred from axioms on a binary relation on 2S 1. 2.3. Decision weight We suppose that there exists a real-valued function v on X, defined up to an increasing linear transformation. v(x) associates to all consequence x a « basic utility » (Loomes and Sugden (1987b)) that measures the psychological value of x. This value: 1) represents the level of satisfaction associated with the fact of having x, 2) is defined independently of choice. In a context of choice, we suppose that there exists a function σ called a weighting function. For any action f, σ associates, to each event A, a number σ f ( A ) between 0 and 1. σ f ( A ) is called a decision weight. Generally speaking, a decision weight depends both on beliefs about events and on consequences of acts. We introduce the following definitions. DEFINITION 1: For all events A, (i) A is (subjectively) unambiguous iff π ∗ (A) = π(A) = π ∗ (A). We denote 2 Sna the set of the unambiguous events. (ii) A is (subjectively) ambiguous iff π ∗ (A) < π ∗ (A). π ∗ (A) - π ∗ (A) measures the subjective ambiguity associated with A. We denote 2 Sa the set of the ambiguous events. DEFINITION 2: A measure of relative satisfaction is a function ξ that associates to all act f and all events A, a real number ξ f A = v ( f A ) − v f , with v f =

∑ π ( A ). v( f A ) . A

ξ f A is a relative increase (if ξ f A ≥ 0) or decrease (if ξ f A ≤ 0) of satisfaction yielded by fA on 2 S .

6

DEFINITION 3: For all act f and all ambiguous event A,

U {B ∈ 2

Sa

}

: f B = f A , where Sa

all events B that belong to the union are disjoint, is called a constant event in 2 for f and is denoted B f A .

(

)

We set: σ f ( A ) = σ ξ f A , π ( A ), π ∗ ( A ), π ∗ ( A ), π ∗ ( B f A ), π ∗ ( B f A ) . This achieves the definition of the decision weight. DEFINITION 4: For all acts and all events, the decision maker is said:

(

(i) Pessimistic iff: ξ f A ≥ ξ g A ⇒ σ ξ f , •

(

A

(ii) Optimistic iff: ξ f A ≥ ξ g A ⇒ σ ξ f , • A

) ≤ σ (ξ ) ≥ σ (ξ

gA

gA

)

,• .

)

,• .

(iii) Uncertainty neutral iff she is pessimistic and optimistic.

We impose the following axioms on σ. C1: ∀A∈ 2 Sa , ∀f∈F, π ∗ (A) ≤ σ f ( A) ≤ π ∗ (A). C2: ∀A,B∈ 2 S , ∀f∈F, π(A) ≥ π(B) ⇒ σ (π ( A ), • C3: ∀A∈ 2

S

(

) ≥ σ (π ( B ), • ) .

)

, ∀f∈F, π ∗ ( B f A ) = π ∗ ( B f A ) or ξ f A = 0 ⇒ σ f ( A) = π(A).

7

C4: For all events A and all act f: (i) If the decision maker is pessimistic then: (i.a) When ξ f A < 0 ,

(π

∗'

)

(

' ' ' ( A ) ≥ π ∗ ( A ) and π ∗ (B f A ) ≥ π ∗ (B f A ) ⇒ σ  π ∗ ( A ), π ∗ ( B f A ), •  ≥ σ π ∗( A ), π ∗( B f A ), •

)

(i.b) When ξ f A >0 ,

(π (A ) ≥ π (A ) and π (B ' ∗

∗

' ∗

fA

)

(

) ≥ π ∗ (B f A ) ⇒ σ π '∗( A ), π '∗( B f A ), •

) ≥ σ (π ( A ), π ( B ∗

∗

fA

), •

)

(ii) If the decision maker is optimistic then: (ii.a) When ξ f A < 0 ,

(π (A ) ≥ π (A ) and π (B ' ∗

∗

' ∗

fA

)

(

) ≥ π ∗ (B f A ) ⇒ σ π '∗ ( A ), π '∗ ( B f A ), •

) ≥ σ (π

∗ ( A ), π ∗ ( B f A

), •

)

(ii.b) When ξ f A >0 ,

(π

∗'

)

(

' ' ' ( A ) ≥ π ∗ ( A ) and π ∗ (B f A ) ≥ π ∗ (B f A ) ⇒ σ  π ∗ ( A ), π ∗ ( B f A ), •  ≥ σ π ∗( A ), π ∗( B f A ), •

)

Under C1, lower and upper probabilities are bounds for the decision weight. Under C2, the decision weight is (weakly) increasing with the probability. Under C3, the decision weight is independent both of events belonging to unambiguous constant events and of ‘irrelevant consequences’. For the first case, consider the act f = xA∪By. Although events A and B are ambiguous, the event A∪B can be unambiguous. This means that the belief about the event “having the consequence x when one chooses f ” is unambiguous. Then, σ f ( A ) = π(A) and σ f (B ) = π(B). For the second case, the decision weight is equal to the probability when the event yields an ‘irrelevant consequence’ that is neither a relative increase of satisfaction nor a relative decrease of satisfaction. Consider the act g that yields x if A or B occur and y if C occurs. Suppose that x yields a relative decrease of satisfaction and that the decision maker is pessimistic. Under C4 (case (i.a)), the decision weight is (weakly) increasing with the upper probability of A (and of B) and with the upper probability of the event “having x if one chooses g ”, that is with B f A = {A, B}.

8

The weighting function has the following properties (all the proofs are given in appendix). PROPERTY 5: ∀A∈ 2 Sna , ∀f∈F, σ f ( A ) = π(A).

PROPERTY 6: For all ambiguous events A : (i) If the decision maker is pessimistic then: (i.a) ξ ( f A ) < 0 ⇒ σ f ( A ) ≥ π(A). (i.b) ξ ( f A ) > 0 ⇒ σ f ( A ) ≤ π(A). (ii) If the decision maker is optimistic then: (ii.a) ξ ( f A ) < 0 ⇒ σ f ( A ) ≤ π(A). (ii.b) ξ ( f A ) >0 ⇒ σ f ( A ) ≥ π(A).

PROPERTY 7: For all events A and all act f: (i) If the decision maker is pessimistic then : (i.a) When ξ f A >0 ,

(π

∗'

)

(

(A ) ≥ π ∗ ( A ) and π ∗ (B f A ) ≥ π ∗ (B f A ) ⇒ σ  π ∗ ( A ), π ∗ ( B f A ), •  ≤ σ π ∗( A ), π ∗( B f A ), • '

'

'

)

(i.b) When ξ f A < 0 ,

(π (A ) ≥ π (A ) and π (B ' ∗

∗

' ∗

fA

)

(

) ≥ π ∗ (B f A ) ⇒ σ π '∗( A ), π '∗( B f A ), •

) ≤ σ (π ( A ), π ( B

fA

), •

)

) ≤ σ (π ( A ), π ( B

fA

), •

)

∗

∗

(ii) If the decision maker is optimistic then: (ii.a) When ξ f A >0 ,

(π (A ) ≥ π (A ) and π (B ' ∗

∗

' ∗

fA

)

(

) ≥ π ∗ (B f A ) ⇒ σ π '∗( A ), π '∗( B f A ), •

∗

∗

(ii.b) When ξ f A < 0 ,

(π

∗'

)

(

(A ) ≥ π ∗ ( A ) and π ∗ (B f A ) ≥ π ∗ (B f A ) ⇒ σ  π ∗ ( A ), π ∗ ( B f A ), •  ≤ σ π ∗( A ), π ∗( B f A ), • '

'

'

)

9

PROPERTY 8: If the decision-maker is uncertainty neutral then σ f ( A ) = π(A).

2.4. From RTc criterion to RTu criterion First, we recall the basic intuition in regret theory. In a context of choice between two acts f and g, the decision-maker compares, for all events A belonging to a constant partition of S, consequence fA and consequence gA. When f is chosen rather than g, the decision-maker feels a regret (respectively a rejoicing) if gA is better than fA, that is if v (g A ) > v ( f A ) (respectively if fA is better than gA, that is if

v ( f A ) > v (g A ) ). Having fA and missing gA is a « composite experience » (Loomes et Sugden (1987a, p.272)). M ( f A , g A ) represents the level of satisfaction derived from this experience, where M(.,.) is a real-valued function on X×X, unique up to a positive linear transformation. The following conditions are imposed on M(.,.). ASSUMPTION 9: ∀x,y∈X, v(x)

>



M(y,x), M(y,z) >M(z,y) and M(x,z) >M(z,x)) ⇒ M(x,z) - M(z,x) > M(x,y) - M(y,x) + M(y,z) - M(z,y). Thus, one supposes that the composite experience felt ex post, that is after the decision and the true event has occurred, is anticipated by the individual. That is, the individual takes into account this potential composite experience at the

10

moment of the decision (ex ante). In RTc, the choice between two acts f and g is then determined by the maximization of the mathematical expectation of the level of satisfaction measured by M(.,.) :

∀f,g∈F, f f g ⇔ ∑ π ( A ).M ( f A , g A ) ≥ ∑ π ( A ). M (g A , f A ) ~ A

(1)

A

where A is an event belonging to a constant partition of S. In RTu, the decision weight that respects C1 to C4 is substituted to the subjective probabilities retained in RTc. So, we have the following decision criterion:

∀f,g∈F, f f g ⇔ ~

∑ σ f ( A ). M ( f A , g A ) ≥ ∑ σ g ( A ). M (g A , f A ) A

(2)

A

where A belongs to a constant partition of S. If the π(A)’s in RTc are interpreted as basic (intuitive) probabilities, RTu is a generalization of RTc: RTc is the special case of RTu either when the subjective ambiguity is null for all events (because of property 5) or when the decision-maker is uncertainty neutral (because of property 8). 3. PROPERTIES OF RTu RTc does not respect transitivity, stochastic dominance and reduction. As a generalization of RTc, RTu also violates these properties (see Villion (2004a)). Contrary to RTc, RTu does not satisfy the sure-thing principle (section 4), the regret-reduction principle and regret-dominance. However, RTu respects under some conditions simple dominance, regret-dominance-under-uncertainty and the identity-regrouping principle.

11

3.1. Simple dominance

{

DEFINITION 12: Let A1 , ... , A j ,. .. , An

} a constant partition of S. A binary relation f~

defined on F satisfies the simple dominance iff:

( ) ( ) for all A ∈ {A ,... , A ,... , A } ] ⇒ f f g.

∀f,g ∈ F, [ v f Aj ≥ v g Aj

j

1

j

n

~

In this section, we suppose that f on F satisfies (2) and we take, on one ~

hand, M(x,y) = M(v(x),v(y)), where M satisfies the assumptions 9 and 10 and is differentiable in v. On the other hand, we suppose that σ verifies C1 to C4 and is differentiable in ξ. PROPOSITION 13: If, for all act f and all events A,

∂M ∂σ . σ f ( A) ≥ − .( 1 − π ( A )). M ( f A , g A ) ∂v ∂ξ

(3)

Then, f verifies the simple dominance. ~

3.2. Regret-dominance-under-uncertainty Quiggin (1990) shows that RTc satisfies a property called regret-dominance. We present in this section a corollary of this result for RTu. DEFINITION 14: Let f and g two acts defined on a constant partition of S

{A ,... , A ,... , A } . f 1

j

m

{

regret-dominates g, denoted f RD g, iff: if there exists n

events Aj∈ A1 ,... , A j ,... , Am events

{

}

such that v ( f Aj ) < v ( g Aj ) , then there exist n

Ak∈ A1 ,... , A j ,... , Am

}

such

that,

for

any

Aj :

v ( g Ak ) ≤ v ( f Aj ) < v ( g Aj ) ≤ v ( f Ak ) .

12

PROPOSITION 15: (Quiggin (1990))

∀f,g∈F, f RD g ⇒

∑ π (A ).M ( f

A

A

, g A ) ≥ ∑ π ( A ).M (g A , f A ) , A

where A is an event that belongs to a constant partition of S and M(.,.) is a realvalued function on X×X that satisfies increasingness. In RTu, because decision weights are not probabilities, regret-dominance is generally not satisfied (see Villion (2004a)). We now define the regret-dominanceunder-uncertainty. DEFINITION 16: Let f and g two acts defined on a constant partition of S

{A ,..., A ,..., A }. f regret-dominates-under-uncertainty g, denoted f RDI g, iff: if 1

j

m

{

there exists n events A j ∈ A1,..., A j ,..., Am

{

exist n events Ak ∈ A1,..., A j ,..., Am

} such that v( f A

} such that, for every A j

) < v( g Aj ), then there

j

:

(i) π ∗ ( Ak ) ≥ π ∗ ( A j ) . (ii) v ( g Ak ) ≤ v ( f Aj ) < v ( g Aj ) ≤ v ( f Ak ) .

PROPOSITION 17: For all acts f and g, If,

(i) f RDI g. (ii) Simple dominance is verified.

Then,

∑ σ f ( A ). M ( f A , g A ) A

≥

∑ σ g ( A ). M ( g A , f A ) A

where A is an event that belongs to a constant partition of S, M is a function that satisfies assumptions 9 and 10 and σ is a weighting function that satisfies C1. 3.3. Identity-regrouping principle Villion (2004a) shows that RTc satisfies a property called the regret-reduction principle. We define in this section a weaker property satisfied by RTu and called the identity-regrouping principle. 13

DEFINITION 18: The probability distribution on X×X induced by π through f and g, denoted πfg , is defined by:

π fg (x , y ) = π ({s ∈ S : f (s ) = x and g (s ) = y }) . Note that π fg ( x , y ) = π gf ( y , x ), ∀( x , y ) ∈ X×X. PROPERTY 19: (regret-reduction principle)

(

)

∀f , g , f ' , g' ∈ F , ∀π fg , π f ' g' , π fg = π f ' g' ⇒ ( f f g ⇔ f ' f g' )

PROPOSITION 20: (Villion (2004a)) All binary relation

f ~

on F that verifies (1) satisfies the regret-reduction principle.

We know that, in RTu, the belief measure is not reduced to a probability measure. Consequently, it is not difficult to show that the model violates the regretreduction principle. An example is given by the event-splitting effect (see tables 3 and 4 below). Nevertheless, RTu respects a weaker principle, that we call the identityregrouping principle. We introduce some definitions before presenting the result. DEFINITION 21: Let f any act. Let presented on

{A1 ,... , Am }

{A1 ,... , Am }

a constant partition of S for f. We say that f is

when, in the presentation of the decision problem,

events Aj and consequences f A j appear explicitly, for all j = 1 to m.

14

DEFINITION 22: Let f an act presented on a constant partition of S {A1 ,... , Am } . Let (l,k) a set of pairs such that 1 ≤ l ≤ k ≤ m. For every pair (l,k) in this set, if f A j = f A j +1 , ∀j = l to k, then we define A' i =

k

U Aj

j =l

where i = 1 to n, n ≤ m, in such

a way that {A'1 ,... , A' n } is a partition of S. We say : (i) {A'1 ,... , A' n } is a partition of S obtained through ordered regrouping of constant events for f in {A1 ,... , Am } ; (ii) f ' presented on

{A'1 ,... , A' n }

and such that f ' A'i = f A'i for all i is an act

obtained through ordered regrouping of constant events for f.

PROPERTY 23: (Identity-regrouping principle) Let f an act presented on a constant partition of S {A1 ,... , Am } . Let f ' presented on a partition of S

{A'1 ,... , A' n }

and obtained through an ordered regrouping of

constant events for f in {A1 ,... , Am } . We have: f ~ f '.

PROPOSITION 24: All binary relation that verifies (2) satisfies the identity-regrouping principle.

4. THE ELLSBERG PARADOX AND EVENT-SPLITTING EFFECTS This section is devoted to two experimental phenomena that are consistent with RTu but not with RTc. 4.1. The Ellsberg paradox

4.1.1 The three-color urn We present the well-known Ellsberg (1961)’s example in table 1.

15

f g f' g'

R 100 0 100 0

B 0 100 0 100 Table 1

Y 0 0 100 100

Events R, B and Y are, respectively: a red, black and yellow ball is thrown in an urn containing 30 red balls and 60 black or yellow balls in unknown proportions. The decision-maker wins 100 USD if she chooses f and a red ball is thrown or if she chooses g and a black ball is thrown. The decision-maker wins 100 USD if she chooses f ’ and a red or yellow ball is thrown or if she chooses g’ and a black or yellow ball is thrown. The preferences generally observed are f g'

fg

and

f f '.

PROPERTY 25: (Sure-thing principle)

(f

A

)

= f A' , g A = g A' , f A C = x , g A C = x , f A' C = y, g A' C = y ⇒ ( f f g ⇔ f ' f g ')

We can see immediately that the preferences revealed by the so-called Ellsberg paradox violate the sure-thing principle. We can also see that RTc respects the sure-thing principle and, consequently, this model cannot predict the paradox. If we apply relation (1) to the Ellsberg example, we have: f

fg

⇔ π(R).M(100,0) + π(N).M(0,100) + π(J).M(0,0) > π(R).M(0,100) + π(N).M(100,0) + π(J).M(0,0) ⇔ [π(R) - π(N)].M(100,0) + [π(R) - π(N)].M(0,100) > 0 ⇔ f'

f g'.

Now, apply RTu (relation (2)): f

fg

⇔ [ σ f ( R ) - σ g ( B ) ].M(100,0) + [ σ f ( B ) - σ g ( R ) ].M(0,100) + [ σ f ( Y ) - σ g ( Y ) ].M(0,0) > 0

(a)

And,

16

g ' f f ' ⇔ [ σ f ' ( R ) - σ g' ( B ) ].M(100,0) + [ σ f ' ( B ) - σ g' ( R ) ].M(0,100) + [σ

f ' (Y )

- σ g' ( Y ) ].M(0,0) > 0

(b)

If the decision-maker is pessimistic, inequalities (a) and (b) can be obtained. In particular, we have the result when we make the not so much restrictive following hypothesis: (i) π(R) = π(B) = π(Y) = 1/3. (ii) π ∗ (R) = π ∗ (R), π ∗ (B) > π ∗ (B), π ∗ (Y) > π ∗ (Y). (iii) π ∗ (B∪Y) = π ∗ (B∪Y), π ∗ (R∪B) > π ∗ (R∪B), π ∗ (R∪Y) > π ∗ (R∪Y). (iv) M(0,0) = 0, M(100,0) >0, M(0,100)< 0, M(100,100) >0. Then, we have 2 : (v) σ f ( R ) > σ g ( B ) under C4 and properties 5 and 7, because ξ f R = ξ g B . (vi) σ f ( B ) = σ g ( R ) under C1 and C3. (vii) σ f ' ( R ) = σ g' ( B ) under C1 and C3. (viii) σ f ' ( B ) > σ g' ( R ) under C4 and properties 5 and 7, because ξ f ' B = ξ g' R . (ix) σ f ' ( Y ) < σ g' ( Y ) under C3 and C4 and property 7, because ξ f 'Y = ξ g'Y . (iv), (v) et (vi) imply f

f g and (iv), (vii), (viii) and (ix) imply g' f f '.

To give an interpretation of the Ellsberg paradox, we take the following numerical example. The beliefs are supposed to be given by: π(R) = π(B) = π(Y) = 1/3, π ∗ (R) = π ∗ (R),

π ∗ (N) = π ∗ (J) = 0,

π ∗ (N) = π ∗ (J) = 2/3,

π ∗ (R∪N) = π ∗ (R∪J) = 1/3,

π ∗ (R∪N) =

π ∗ (R∪J) = 1, π ∗ (N∪J) = π ∗ (N∪J) = 2/3.

17

For all events A and all act f, the decision weight σ f ( A ) is defined by: π ( A ), ∀A ∈ 2 Sna σ f ( A) =  Sa ∗ λ f ( A ). π ∗ ( A ) + 1 − λ f ( A ) . π ( A ), ∀A ∈ 2

(

)

(4)

where λ is a function that associates to every event A and every act f a value

λ f ( A ) in [0,1]. λ is such that, for all ambiguous events A and all act f: (i) λ is monotonic and differentiable in ξ f A , π(A), π ∗ (A), π ∗ (A), π ∗ ( B f A ) and π ∗ ( B f A ). (ii) ∂λ

∂π < 0 .

(

)

(iii) ξ f A = 0 or π ∗ (B f A ) = π ∗ (B f A ) ⇒ λ f ( A ) =

π ∗(A ) − π (A ) . π ∗(A ) − π ∗(A )

(iv) (iv.a) If ∂λ

∂ξ > 0 then:

∂λ ∂λ  ∗ 1 − λ f (A ) > ∂π ∗ ( A ) . π ( A ) − π ∗ ( A ) and ∂π ∗ (B ) < 0 fA   < 0,  ∂λ ∂λ >0 . π ∗ ( A ) − π ∗ ( A ) and λ f (A ) <  ( ) ( ) ∂π A ∂π B ∗ ∗ f A 

(

- When ξ f A

(

)

)

∂λ ∂λ  ∗ 1 − λ f (A ) < ∂π ∗ ( A ) . π ( A ) − π ∗ ( A ) and ∂π ∗ (B ) > 0 fA   >0,  ∂λ ∂λ  ( ) ( ) ∂π A ∂π B ∗ ∗ f A 

(

- When ξ f A

(iv.b) If ∂λ

(

)

)

∂ξ < 0 then:

∂λ ∂λ  ∗ 1 − λ f ( A ) < ∂π ∗ (A ) . π ( A ) − π ∗ ( A ) and ∂π ∗ (B ) > 0 fA   < 0,  ∂λ ∂λ . π ∗ ( A ) − π ∗ ( A ) and  ( ) ( ) ∂π A ∂π B ∗ ∗ f A 

(

- When ξ f A

(

)

)

18

∂λ ∂λ  ∗ − > − 0,  ∂λ ∂λ >0 . π ∗ ( A ) − π ∗ ( A ) and λ f (A ) <  ∂π ∗ (A ) ∂π ∗ (B f A ) 

(

- When ξ f A

(iv.c) If ∂λ

∂ξ

)

(

= 0 then λ f ( A ) =

)

π ∗ ( A) − π ( A) . π ∗ ( A) − π ∗ ( A)

The decision weight defined in this way verifies C1, C2 (by (ii)), C3 (by (iii)) and C4 (by (iv)). Note that if ∂λ

∂ξ is strictly positive, strictly negative or null then

the decision-maker is, respectively, pessimistic, optimistic or uncertainty neutral. As an example, a function λ that satisfies conditions (i) to (iv) is given by 3 : ∀A∈ 2 Sa , ∀f∈F,

  π ∗ (A ) − π (A )      ∗ − π A π A ( ) ( )    ∗ λ f ( A ) = 12 tanh a. π ∗ (B f A ) − π ∗ (B f A ) .ξ f A + 12 ln + ∗ π ( A ) − π (A )     1 − ∗   π ( A ) − π ∗ (A )    

(

)

(5)

1 2

where a is a real number that is strictly positive, strictly negative or null when the decision-maker is, respectively, pessimistic, optimistic or uncertainty neutral. We suppose that preferences on acts verify (2) where v(x) = x, ∀x∈X, for all act f

and

all

event

A,

σ f ( A)

is

given

by

(4)

and

(5),

and

M(v( f A ),v( g A )) = v ( f A ) + 1 − b v ( f A ) − v ( g A ) , with 0 ≤ b ≤ 1. If we take, for example, a=0.01 et b=0.99, we have (lines represent acts and columns represent events): Matrix of the σ f ( A ) 1 / 3 1 / 3  1 / 3  1 / 3

1/3 1/3 0.194 0.406  0.472 0.260   1/3 1/3

Matrix of the M ( f A , g A ) . 100.63 −1732  −1732 100.63  . 100.63 −1732 .  . 100.63  −1732

   100   100  0 0

19

Consequently, RTu predicts f

f g and g ' f f ' 4. This result is explained by the

characteristics of the beliefs and by the attitude of decision-maker faced with uncertainty: 1) We have

σ h ( R ) = 1 / 3 , ∀h ∈ {f , f ' , g , g '} , σ f ( N ) = σ f ( J ) = 1 / 3

and

σ g' ( N ) = σ g' ( J ) = 1 / 3 because the events « to throw a red ball » and « to throw a black or yellow ball » are felt unambiguous. 2) The events « to throw a red or yellow ball », « to throw a black ball » and « to throw a yellow ball » are felt ambiguous. Then, the decision weights σ g ( J ) and

σ f ' ( N ) are high because they are associated with relative decreases of satisfaction ( ξ g J and ξ f ' are negative). And, the decision weights σ g ( N ) and σ N

f'(J )

are low

because they are associated with relative increases of satisfaction. 4.1.2 The two-color urns In the other example given by Ellsberg (1961), one has two urns. In the first one, there are 100 red or black balls, in unknown proportions. In the second one, there are 50 red balls and 50 black balls. In the two cases, the decision-maker wins 100 USD if she bets on red (respectively on black) and a red (respectively black) ball is thrown in the urn. The decision problem can be presented in a unique decision matrix (table 2) where event RR is « a red ball is thrown in the first urn and a red ball is thrown in the second urn ». Events RN, NR and NN are defined in the same way. The preferences generally observed are g RR f g f' g'

100 100 0 0

RN

100 0 0 100 Table 2

f f and g ' f f

'.

NR

NN

0 100 100 0

0 0 100 100

Let us apply RTu.

20

Suppose that the beliefs are such that: (i)

π(RR) = π(RN) = π(NR) = π(NN) = 0.25.

(ii)

π ∗ (RR∪NR) = π ∗ (RR∪NR) = 0.5 = π ∗ (RN∪NN) = π ∗ (RN∪NN).

(iii)

π ∗ (RR), π ∗ (RN), π ∗ (NR) and π ∗ (NN) are strictly lower than 0.25 — take 0 for example.

(iv)

π ∗ (RR), π ∗ (RN), π ∗ (NR) and π ∗ (NN) are strictly upper than 0.25 — take 0.75 for example.

Recall that π ∗ (RR∪RN), π ∗ (NR∪NN), π ∗ (RR∪RN) and π ∗ (NR∪NN) have to satisfy B2 to B4 — take π ∗ (NR∪NN) π ∗ (RR∪RN) = π ∗ (NR∪NN) = 0, π ∗ (RR∪RN) =

π ∗ (NR∪NN) = 1 for example. Then, it is not difficult to show that, under C4 and property 7, we have:

σ f ( RR ) < σ g ( RR ) , σ f ( NN ) > σ g ( NN ) , σ f ( NR ) < σ g ( RN ) and σ g ( NR ) < σ f ( RN ) . If we suppose that M(100,100) >0, M(0,0)≤ 0, M(0,100)< 0 and M(100,0) >0, then RTu predicts g

f f. In the same way, RTu predicts g ' f f ' 5. 4.2.Event-splitting effects

Consider the tables 3 and 4.

f g

where

A1 x z

A2 t y Table 3

π(Ai )=1/3,

∀i=1,2,3,

A3 t y

π ∗ (B) < π ∗ (B),

f' g'

∀B∈ 2 S ,

A1 A2 ∪A3 x t z y Table 4 x,y,z,t

are

monetary

consequences such that x >y >t,z ≥0. If the preferences satisfy the regret-reduction principle, then we have f f g iff f ' f g' . In particular, we have this relation if RTc represents the preferences.

21

However, experimental results suggest that, for a significant part of subjects, we have g f f and f ' f g' (Starmer and Sugden (1993), Humphrey (1995)). This is a form of event-splitting effect (ESE), by which the subjective weight given to an outcome depends on the number of presented events in which it occurs, as well as on their combined probability. If RTu represents the preferences, then,

[

]

g f f ⇔ σ g ( A1 ). M (z , x ) + σ g ( A2 ) + σ g ( A3 ) . M (y , t ) >

[

]

σ f ( A1 ). M (z , z ) + σ f ( A2 ) + σ f ( A3 ) . M (t , y )

f ' f g' ⇔ σ f ' ( A1 ). M (x , z ) + σ

f'

( A2

(a)

∪ A3 ). M (t , y ) >

σ g' ( A1 ). M (z , x ) + σ g' ( A2 ∪ A3 ). M (y , t )

(b)

(a) et (b) ⇒

[σ

g

( A2 ) + σ g ( A3 ) − σ g ( A2

[

]

]

∪ A3 ) . M (y , t ) > σ f (A2 ) + σ f (A3 ) − σ f (A2 ∪ A3 ) .M (t , y )

by addition and because σ f (B ) = σ

(c), f'

(B ) , σ g (B ) = σ g' (B )

for all event B.

Inequality (c) is not inconsistent with RTu. For example, a weighting function given by relations (4) and (5) is superadditive

(subadditive)

 that

is

such

that

σ ( π ( A ∪ B ), •) > σ ( π ( A ), •) + σ ( π ( B ), •) ( σ ( π ( A ∪ B ), •) < σ ( π ( A ), •) + σ ( π ( B ), •) )  for consequences that lead to relative increases (decreases) of satisfaction. In that case, inequality (3) can be satisfied if M(y,t) >0 and M(t,y)< 0. Then, RTu can predict ESEs when the subjective ambiguity is non-null. We now present two models that can predict ESEs. We show that, contrary to RTu, these models do not respect the identity-regrouping principle.

22

Starmer and Sugden (1993) show that a modified version of prospect theory (Kahneman and Tversky (1979)) predicts ESEs. This modified version is called « stripped-down prospect theory ». In Kahneman and Tversky (1979)’s theory, there is an editing phase and one of the editing operations is that of combination. This is the operation of « combining the probabilities associated with identical outcomes ». After the editing phase, a « regular lottery » (x , p x ; t , pt ) , with x ≥ 0 ≥ t, is evaluated by: σ ( p x ).v (x ) + σ ( pt ).v (t ) . v( • ) is a real-valued increasing function on XM unique up to multiplication by a positive constant, with v(0) = 0 at the reference point. σ( • ) is a weighting function that transforms (objective) probabilities. σ( • ) is subadditive for small probabilities, so

that

if

p(A2 )

and

p(A3 )

are

two

small

probabilities,

σ(p(A2 ∪ A3 )) < σ(p(A2 )) + σ(p(A3 )). Stripped-down prospect theory is prospect theory without editing phase. Apply it to decision problems in tables 3 and 4. If t = 0, g

ff

and f '

f g'

is obtained

when [σ ( p( A2 ) + σ ( p( A3 )].v (y ) > σ ( p( A2 ∪ A3 ).v (y ) . This inequality is verified if σ( • ) is subadditive in the relevant range. Under the same conditions, stripped-down prospect theory also predicts f ~ f '

and g

f g'.

But, g’ is obtained through an

ordered regrouping of constant events for g. Then, g

f g'

is a form of ESE that

violates the identity-regrouping principle.

Humphrey (1998) proposes two modifications of Neilson (1992)’s model. For all lottery p defined on a set of monetary consequences {x1,...,xm} , (i) n(p) is the number of consequences in p with strictly positive probability and, (ii) rp is the reference wealth level, with: r p =

Min x i i

if n(p) > 1 and rp = 0 if n(p) = 1, where the

xi are consequences to which p assigns positive probability. All lotteries p defined on

23

{x1,...,xm} are evaluated by

∑ p .u i

s( p )

(x i ) where u s ( p ) is a utility function and s(p)

i

is : (i) The number of consequences xi > rp listed in the lottery to which p assigns positive probability (model 1). (ii) The number of consequences xi = rp

listed in the lottery to which p

assigns positive probability (model 2). Two alternative hypotheses are defined: - The

frequency

effect

hypothesis

(FEH)

stipulates:

u1 (x) < ...< um (x), ∀x > 0, u1 (0) = ...= um (0) and u1 (x) > ...> um (x), ∀x < 0. - The

boundary

effect

hypothesis

(BEH)

stipulates:

u1 (x) > ...> um (x), ∀x > 0, u1 (0) = ...= um (0) and u1 (x) < ...< um (x), ∀x < 0. Model 1 + FEH

represent

a

preference

for

frequency

of

successes.

Model 2 + BEH represent an aversion to frequency of failures. Model 1 + FEH and Model 2 + BEH can predict g predict f ~ f ' and g

f g'.

ff

and f '

f g'

in tables 3 and 4. Model 1 + FEH also

Model 2 + BEH also predict f '

ff

and g ~ g'. But, g’ (f ’) is

obtained through an ordered regrouping of constant events for g (f). Then, g and f '

ff

f g'

are forms of ESE that violate the identity-regrouping principle. 5. EXPERIMENT

Villion (2000)

provides

experimental

evidence

about

regret

and

disappointment (as defined in Loomes and Sugden (1986)) effects. Two uncertainty contexts are investigated: a risky context (as in Loomes and Sugden (1987) and Loomes (1988) experiments) and an objective ambiguity context. More precisely, uncertainty is about the color of a ball thrown in an urn. And, we say that we are in a risky context when the proportion of balls of each color is precisely known. When this proportion is not precisely known for each color, we say that we are in an objective ambiguity context. The urns are constructed in such way that, if we make 24

an insufficient reason hypothesis, the choice problems in the risky context and the choice problems in the objective ambiguity context are equivalent. First, if regret theory is valid and if the subjects apply an insufficient reason principle, regret effects should be the same in the two contexts. The results show that regret effects are not independent of the uncertainty context. This contradicts both RTc and RTu. Second, if disappointment theory (Loomes and Sugden (1986)) is valid and if the subjects apply the insufficient reason principle then, the disappointment effects should be the same in the two contexts. The results show that it is not the case and support, on the contrary, a pessimism hypothesis as the one defined in RTu. In particular, there is one pair of choices for which we can clearly test RTu predictions. This pair appears in table 5. In urn 1, there are 60 yellow balls and 40 black balls: we are in a risky context. In urn 2, there are 45 yellow balls, 25 black balls and 30 yellow or black balls in unknown proportions: we are in an objective ambiguity context. Y1 (B1)

is the event: « a yellow (black) ball is thrown in the

urn 1 ». Consequences are expressed in FRF. Urn 1 f1 g1

Urn 2

Y1

B1

0 700

2300 700

f2 g2

Y2

B2

0 700

2300 700

Table 5 Consider the predictions of RTu. Suppose that individuals apply an insufficient reason principle and that they consider subjective ambiguities for events Y1 and B1 less than for events Y2 and B2. That is, π(Y1) = π(Y2) = 0.6, π(B1) = π(B2) = 0.4, π ∗ (B2) - π ∗ (B2) > π ∗ (B1) - π ∗ (B1) and

π ∗ (Y2) - π ∗ (Y2) > π ∗ (Y1) - π ∗ (Y1) 6. If preferences satisfy (2), then f1

f g1

and g2

f f2

imply:

25

[

]

[

]

M (0,700 ). σ f 1 (Y1 ) − σ f 2 (Y2 ) + M (2300,700 ) σ f 1 (B1 ) − σ f 2 (B 2 ) > 0 because, under C3,

σ g (Y1 ) = σ g (Y2 ) 1

2

and

σ g (B1 ) = σ g (B 2 ) . 1

2

If M(0,700)< 0 and M(2300,700) >0, then we have the required inequality because, under C4 (and property 7), we have, for a pessimistic individual:

σ f (Y1 ) < σ f (Y2 ) and σ f (B1 ) > σ f (B 2 ) . 1

2

1

2

Experimental results are clearly in accordance with RTu predictions. Denote f1 / g1 the ratio number of subjects that have chosen f1 / number of subjects that have chosen g1 . For the four groups of subjects, we have systematically f1 / g1

above f2 / g2 , and the difference between the two ratio is

statistically significant for one group (α = 0.025). If we combine the data for all groups, we have 53 subjects that have chosen f1 and 50 subjects that have chosen g1 . And, we have 33 subjects that have chosen f2

and 70

subjects that have

chosen g2 . The ratio f1 / g1 is statistically above the ratio f2 / g2 (α = 0.005). So, there is clearly a tendency to switch from f1

f g1

to g2

f f2 .

Moreover, among the

103 subjects, only 8 of them have chosen g1 and f2 . In others words, 95 subjects among 103 have made choices in accordance with RTu. REMARK 26:

Consider

the

following

alternative

explanation

of

the

above

experimental results 7. Suppose

that

individuals

are

not pessimistic

but

neglect ambiguous

information in such a way that they assign the following decision weights:

σ(Y1) = f1

60 60 + 40

f g1

, σ(B1) =

and

g2

40 60 + 40

f f2

, σ(Y2) =

are

45 25 + 45

predicted,

and σ(B2) = for

25 25 + 45

example,

. Then the preferences

by

the

simple

model:

V ( f ) = ∑ σ ( A ).v ( f A ) , ∀A∈2S, ∀f∈F 8. A

26

6. DISCUSSION We have presented a generalization of regret theory (RTc) based on an intuitive view of beliefs and on pessimism. The model obtained (RTu) provides and explanation of the Ellsberg paradox and of event-splitting effects. As RTc, RTu violates transitivity, stochastic dominance and the reduction principle. RTu violates also the sure-thing principle, regret-dominance and regretreduction. However, RTu satisfies simple dominance and regret-dominance-underuncertainty under some conditions and respects the identity-regrouping principle. At this point, we can conclude that RTu is a descriptive model that explains most of the known experimental phenomena, in the limits given by the properties respected by the model. At the methodological level, RTu adopts a non-conventional approach because the model implies the use of introspective methods in experiments. Models that give place to introspection are extremely rare in economics. The rejection of introspection dates back to early in twentieth century when cardinal utility — in a Benthamite or Bernouillian psychological sense — gave way to ordinal utility. Allais (1991) precises: « Today, given the positions taken by some eminent economists which, with some rare exceptions, are as spectacular as they are dogmatic, an intolerant orthodoxy has banished, almost totally, cardinal utility, and, in general, any psychological introspection from economic science. » (Allais (1991, p.104)). As for the representation of tastes, the representation of beliefs in economics generally excludes the use of introspection: decision-oriented views dominate, while intuitive views are quasi absent. The reasons why introspection tends to be excluded are linked to the roles that concepts of choice and preferences play in traditional economic analysis. The main arguments are: 1) axiomatization of one binary relation is the unique satisfactory representation of preferences; 2) choices are the unique precisely and 27

objectively observable data that can reveal the preferences; 3) choices provide a sufficient knowledge of the decision-maker preferences 9. However, Allais (1953), Ellsberg (1961), Lichtenstein and Slovic (1971) or Kahneman and Tversky (1979,1982) have opened a way that challenges the above arguments. While they stimulated the development of experimental methods, they initiated researches toward a better understanding of the cognitive processes of decision under uncertainty 10. Consequently, to consider that these works have some interest for economics is to consider that all the data that give an answer to the question « How people make decision? » are relevant. To understand how people make decisions, we have not to put a frontier between introspective methods and “extrospective” ones. In particular, there is an essential limitation when one wants to resume the preferences of the decision-maker solely through a binary relation and to reveal these preferences through choices. In doing so, one cannot achieve the separation between the representation of (pure) tastes and the representation of (pure) beliefs. This is an essential limitation not only from a purely descriptive point of view but also for the predictive power of the theory (see Villion (2004b)). Another difficulty with the traditional approach emerges when we deal with normative questions. Models that try to represent the preferences uniquely through an axiomatization of a binary relation reduce rationality to the consistency of choice (see Sen (1985)). But, it turns out that expected utility is the best model to represent consistent choices. Consequently, axiomatic models that are more general than expected utility tend to be not admitted on normative grounds (see, for example, Sarin (1992)). To consider rational choice out of expected utility, we have to deal with decision consistency, that is consistency of the mental process leading to choices. A rational decision is a decision of an individual who can justify his choices and who acts correctly to achieve his ends, these ones being « not 28

themselves given by reason » (Sugden (1991, p.753) 11. For example, if regret aversion is a psychological characteristic of the decision-maker, it is rational for him to take into account in decision the possibility of feeling regret after the uncertainty is resolved. Similarly, if pessimism is a psychological characteristic of the decision-maker and if beliefs are given by uncertain subjective probabilities, it is rational to affect decision weights to consequences of acts that take into account pessimism. Here again, choice observation is insufficient. Introspection is necessary to discriminate between an irrational decision and a rational one. APPENDIX: PROOF Proof of Property 5: Apply B5 and C1.

Proof of Property 6: The result follows immediately definition 4.1.4 and axiom C3.

Proof of Property 7: The non-ambiguous case is obvious. Let A∈ 2 Sa and f∈F. Consider part (i.a) and case π ∗ (A) in property 4.1.3. We have ξ f A >0 and set down σ f ( A ) = π(A) - ε, with ε >0, which respects property 4.1.2 (part (i.b)), '

Let π ∗ ( A ) > π ∗ (A).

29

(

' Suppose, contrary to property 4.1.3 (part (i.a)), that σ  π ∗ ( A ), •  > σ π ∗ ( A ), •

with

(

σ  π ∗ ( A ), •  − σ π ∗( A ), '

•

) > ε.

Then,

we

σ f ( A) >π(A),

have

)

which

contradicts property 4.1.2 (part (i.b)). The other case and parts (i.b), (ii.a) and (ii.b) can be verified in the same way.

Proof of Property 8: By definition, an uncertainty neutral decision-maker is pessimistic and optimistic. •

Consider the case ξ f A < 0.

Under C4 (i.a) and Property 7 (ii.b),

(π

∗'

)

'

(

'

'

) (

)

(A ) ≥ π ∗ ( A ) and π ∗ (B f A ) ≥ π ∗ (B f A ) ⇒ σ π ∗ ( A),π ∗ ( B f A ),• = σ π ∗ ( A),π ∗ ( B f A ),• (a)

Under C4 (ii.a) and Property 7 (i.b),

(π (A ) ≥ π (A ) and π (B ' ∗

•

∗

' ∗

fA

)

(

) (

)

) ≥ π ∗ (B f A ) ⇒ σ π ∗' ( A), π ∗' ( B f A ),• = σ π ∗ ( A), π ∗ ( B f A ),• (b)

Consider the case ξ f A >0.

Under C4 (ii.b) and Property 7 (i.a),

(π

∗'

)

'

(

'

'

) (

)

(A ) ≥ π ∗ ( A ) and π ∗ (B f A ) ≥ π ∗ (B f A ) ⇒ σ π ∗ ( A),π ∗ ( B f A ),• = σ π ∗ ( A),π ∗ ( B f A ),• (c)

Under C4 (i.b) and Property 7 (ii.a),

(π (A ) ≥ π (A ) and π (B ' ∗

•

∗

' ∗

fA

)

(

) (

)

) ≥ π ∗ (B f A ) ⇒ σ π ∗' ( A), π ∗' ( B f A ),• = σ π ∗ ( A), π ∗ ( B f A ),• (d)

By definition of σ f ( A ) , (a), (b), (c), (d), C3 (for the case ξ f A =0) give the result.

Proof of Proposition 13: Let f and g two acts.

{

Let A1 , ... , A j ,. .. , An

} a constant partition of S. 30

•

We have to show that:

If, ∂M ∂σ .σ f ( Aj ) ≥ − .( 1 − π ( A j )). M ( f A j , g A j ) for all A j in A1 ,... , A j ,... , An ∂v ∂ξ Then,

{

( ) ( ) for all A ∈ {A ,... , A ,... , A } ] ⇒ ∑ σ ( A ). M ( f

[ v f Aj ≥ v g Aj

j

1

j

Aj

•

{

f

n

j

Aj

)

, g Aj ≥

∑ σ g ( A j ). M Aj

(g

Aj

, f Aj

}

)

(a)

}

Let A j ∈ A1 ,... , A j ,... , An .

The relation:

( ) ( )⇒σ

v f Aj ≥ v g Aj

f

(

)

(

( A j ). M f Aj , g Aj ≥ σ g ( A j ). M g Aj , f Aj

)

is satisfied iff: ∂σ ∂M .M( f A j ,gA j ) + .σ f ( Aj ) ≥ 0 ∂v ∂v ⇔ •

∂M ∂σ .σ f ( Aj ) ≥ − .( 1 − π ( A j )). M ( f A j , g A j ) ∂v ∂ξ

(b)

If relation (b) is satisfied for all A j then relation (a) is satisfied.

Proof of Proposition 17:

{

}

Let f and g two acts defined on a constant partition of S A1 ,... , A j ,... , Am . Let, for all x and y in X, M(x,y) a function that satisfies assumptions 9 and 10. • Suppose, without any lost of generality, that v ( f Aj ) < v ( g Aj ) , ∀j∈ {1,... ,l } and v ( g A j ) ≤ v ( f A j ) , ∀j∈ {l + 1,... , m} . If f RDI g then l < pair, if l ≥

m 2

m 2

if m is pair and l