choice under risk and the security factor: an axiomatic model

J E A N - Y V E S J A F F R A Y

C H O I C E U N D E R RISK AND THE S E C U R I T Y F A C T O R :

AN A X I O M A T I C M O D E L

ABSTRACT. The particular attention paid by decision makers to the security level ensured by each decision under risk, which is responsible for the certainty effect, can be taken into account by weakening the independence and continuity axioms of expected utility theory. In the resulting model, preferences depend on: (i) the security level, (ii) the expected utility, offered by each decision. Choices are partially determined by security level comparison and completed by the maximization of a function, which express the existing tradeoffs between expected utility and security level, and is, at a given security level, an affine function of the expected utility. In the model, risk neutrality at a given security level implies risk aversion.

Keywords: decision theory, risk, expected utility, security level, risk aversion.

1. I N T R O D U C T I O N

Since it was developed by von Neumann and Morgenstern (1944), expected utility (EU) theory has been the standard model of decision making under risk, for descriptive as well as normative purposes. Evidence of its deficiencies as a descriptive model has however gradually accumulated to such an extent that doubt has finally been cast also on its validity as a normative model.

This has motivated proposals by, among others, Kahneman and Tvers- ky (1979), Chew (1981), Machina (1982), Quiggin (1982), Segal (1984), and Yaari 0987) of alternative axiomatic models in order to account for the systematic deviations from EU theory reported by experimenters - in particular the violations of the independence axiom first revealed by Allais' (1952/1979) famous experiment. This last concern necessitates a weakening of the independence axiom; its degree of relaxation is however extremely variable from one model to another, and the justification of the particular version of the axiom used is not always clear. Yet, as the axioms of EU theory derive from extremely appealing rationality prin- ciples, it would seem desirable that the axioms of an alternative model remain as close to the original ones as possible, and that clear justification

Theory and Decision 24 (1988) 169-200. �9 1988 by D. Reidel Publishing Company.

170 JEAN-YVES JAFFRAY

be given concerning any restrictions made. This paper is devoted to the development of an axiomatic model along these lines.

Our starting point will be the observation that the security factor (Lopes, 1986) - the desire for security - appears to play a fundamental

part in the determination of people's choicesJ It makes people attach

special importance to the worst outcomes of risky decisions as well as to

the sole outcomes of riskless decisions; this later phenomenon is known as the certainty effect (Kahneman and Tversky, 1979) or complementary effect (AUais, 1952/1979).

We shall thus reexamine the classical justifications of EU theory

rationality axioms and specify the restrictions that have to be placed on them to allow for a possible effect of the security factor. As we shall see, both the independence axiom and the continuity axiom must be weaken-

ed, which, in turn, makes the addition of a dominance axiom necessary. The axiom system once specified, we shall look into the main features

of the model. It will first be shown that EU theory remains valid among

decisions offering the same security level. Tradeoffs between expected utility level and security level - where they exist - will then be shown to

take a specific form, and preferences complying with the model will be characterized through a representation theorem (subsection 4.4.2). Parti- cular forms of the model, corresponding to stronger independence re-

quirements, will also be described. Il will moreover be shown that the model is able to explain risk-aversion

by the security motivation alone. We shall also examine to what extent

it accounts for the paradoxes of EU theory. Finally the two models will be compared from the practical point of view by considering the assess-

ments they require and the calculations they involve in applications.

2. THE MODEL

2.1. Lotteries

L e t ( b e a set of outcomes and .~be the set of all lotteries o n ~ i.e., of all probability distributions o n ( w h i c h have a finite support. In particular, any degenerate distribution yielding an outcome c with probability one belongs to .~, and is denoted by Ic; if necessary, we will identify the set of degenerate distributions with

CHOICE UNDER RISK AND THE SECURITY FACTOR 171

J is a convex set (in the real vector space of measures with finite support on ~) ; the convex linear combination (c.l.c.) of lotteries ~ in_--I )~iPi, (i.e., the combination of lotteries Pi with non-negative coeffi- cients s adding up to one), is the lottery which yields any outcome c with probability ~n= 1 ~,iPi(Icl). In particular, ~n= 1 Pilci is the lottery with support [ci, i=1 ..... n I that yields outcome ci with probability Pi.

The mathematical expectation of function f with respect to lottery P is denoted by Epf.

2.2. Weak Ordering Assumption

Let preferences under risk be characterized by a binary relation, ~ , in ~ ; to relation ~ are standardly associated its asymmetric part, ),-, its symmetric part, ,~, as well as the opposite relations, ~ and -~, of ~ and >-.

Whenever relation ~ is transitive and complete (hence also reflexive) it is a weak order, and (~, ~ ) is a weakly ordered set. Our first axiom is:

(A1) ('~, ~ ) is a weakly ordered set.

2.3. Security Levels

The restriction of preference relation ~ to degenerate lotteries can be considered as a relation in ~ characterizing preferences under certainty. Under A1, ~ ~ ) is itself a weakly ordered set.

Order intervals [c ~ ( : ) ~ ' c,~ c ] and [c ~ : c' ~ c ~ c'], where c' >- c", will conveniently be denoted by [c', ~ ] and [c", c'], respectively.

To each lottery P can then be associated an outcome cp, called a minimal outcome of P, satisfying P([cp])> 0 and P([cp,--*]) = 1. Since ~ ) is not linearly ordered in general (i.e., ~ is not an antisymmetric relation in ( ) , a lottery may have several minimal outcomes; however, if c~ is another minimal outcome of P, necessarily, c'v-cp, i.e., both belong to a same equivalence class, which characterizes the security level ensured by P; to refer to a security level, we shall simply quote one of its members.

172 J E A N - Y V E S J A F F R A Y

2.4. Independence Assumption

The independence axiom of EU theory states that, given lotteries P, Q and R and real number ~., with 0<~.~< 1,

P ~ Q if and only if ~,P+ (1 -~,)R ~ ~,Q+ (1 -A)R.

The usual argument for defending this axiom introduces a two stage process of realization of the lotteries (see Figure 1), which transforms the above statement into the following consistency requirement: decision 8' should be preferred to decision 8" in the first decision tree if and only if decision d' is preferred to decision d" in the second one.

ProME) =), Prob (Ee)= 1 -2

~ P

q ~ P

R ~ E~q R

decision tree # l decision tree # 2 (decision node: NK; security levels: (decision node: N2; security levels: cp for 8', c O for 8") ~f~ [c~ cRI for d',~lnf leo, cR] for d")

t

at Ni, same security level for 6' and 8"; at N:, same security level for d' and d" ]

Fig. 1. Justification of assumption A2(i).

The argument is that, if event E occurs, d' (resp. d") offers the same lottery as 8' (resp. 8"), whereas, if event E c occurs, the choice made is immaterial.

However it may happen that d' and d" offer a common security level, whereas 8' and 8" ensure different ones (such as, e.g., in Allais' (1952/1979) example), so that decision makers who pay particular attention to security levels do not necessarily display the expected consistency in their choice. For this reason, we shall only require a weaker independence property:


(A2) (i) / f P , Q, and R belong to ~, Cp~CQ, and 0<Z~< 1, then P ~ Q if, and only if, ~,P+ (1 -~,) R ~ AQ+(1 - ) 0 R.

(ii) I f P', P", Q', and Q" belong to .~ Cp, ~ co,, Cp,, ~ CQ,,, and 0<~)~ <~1, then P' ~ P" and Q' ~ Q" imply that A P' + (1 - A) Q' ~ A P " + ( 1 - s and, i f moreover P '>-P" and ; t>0, then AP'+(1-~ , ) Q' >- ~ ,P"+(1 -Z) Q".

Thus A2 (i) restricts the standard independence assumption of EU theory to decision situations where security motivation cannot come into play. On the contrary, A2 (ii) (which is implied by the axioms of EU theory) applies to situations where it may be brought into play. The disparity in security levels between lotteries AP'+ (1 -~,) Q' and ~,P" + (1 - Z ) Q" is however the same as that existing between lotteries P ' and P " or Q' and Q"; thus, in the two-stage interpretation (see Figure 2), A2 (ii) exactly requires that the decision maker not care whether Nature's choice of E or E c takes place before or after his own choice, as long as the security levels involved remairi the same. Thus A2 in fact requires that there be no limitations to the independence principle, except those imposed by security level considerations.

Prob(E) =,~ Prob (E ' )= I - , l

6 ~ q'

N2 d ' ~ ~ E Ec P '

d " ~ Q"

decision tree # I decision tree # 2 (decision nodes: Ni, ~1; security levels: (decision node: Nz; security levels: at N h cp. for 8', cp for 6"; ~ f Ic~, % I for d', I~f Ic~, %.1 for d"} at Ni, cq. for ~', cq for ~") ~

] same pair of security levels [

at N, for (c~', ~"); at N~ for (~', ~'); at N z for (d, d') ]

Fig. 2. Justification of assumption A2 (ii).


2.5. Continuity Assumption

If decision makers pay particular attention to security levels, lottery ~,P+ (1 -~.) Q does not necessarily become "psychologically close" to P when 3. tends to one (resp. to Q when ~. tends to zero) since, when the security levels offered by P and Q differ, that of ).P+ (l -~,) Q remains constant and equal to the smaller of the two. For this reason, we shall restrict our continuity requirement as follows:

(A3) I f P, Q, and R belong to J , Cp ~ CQ and P ~- R ~- Q, there exist 2. andg, with 0 < s 1 and O<t~ < 1, such that s (1 -~ ) Q ~'R >-/~P+ (1-U) Q.

2.6. Dominance Assumption

Let us denote by ~ 9 the (first order) stochastic dominance ordering, which is defined in ~ by

p ~ o Q if, and only if, for all c in ~ P([c, ~]) i> Q([c, -,]).

Thus P--D Q if and only if P and Q are of the form

P= ~ ~i Ic'i" Q= ~ Ai let, with Ic, i - Ic, I for i= 1 . . . . . n, i = l I = l

and it follows from AI and A2(i) by an inductive argument that p n Q implies that P - Q .

The possibility that P ~-OQ and P ~ Q simultaneously hold has however not been excluded so far by the axioms (as shown by ,~3 of 4.1); we thus need a dominance axiom:

(A4) I f P and Q belong to ~,, and P ~- DQ, then P ~- Q.

Please note that, from now on, it is assumed that preferences satisfy axioms A1 to A4.

2.7. Utifity Functions

Before beginning to study the properties of the model, let us recall some important definitions and properties: A real function, f , defined on ~ i s a utility on weakly ordered set (;~., ~) whenever, for x, y in ~J~,,f(x)~f(y)

C H O I C E U N D E R R I S K A N D T H E S E C U R I T Y F A C T O R 175

if and only if x ~ y; f is then said to represent ~ . If ~ ' i s a convex set and utility f moreover satisfies

f(~,x+(1-J~)y)=),f(x)+(l-)Of(y), for all x, y in ~ ~ 0~<~,~<1,

f i s a linear utility. When a linear utility exists, it is unique up to a positive affine transformation, i.e., any other linear utility, g, has the form af+[3, with a > 0; thus if Xo >-xl and ko > kl, conditions g(xo)= ko and g(xl)= kl specify one particular linear utility.

3. E X P E C T E D U T I L I T Y R E P R E S E N T A T I O N F O R P R E F E R E N C E S AT A G I V E N S E C U R I T Y L E V E L

For any Co inC let ~Jco= IPE~: cp-col; thus ~J/co is the set of all lotteries ensuring security level c o .

3.1. Linear Utility Representation for Preferences in ~Jco

It is straightforward that --@co is a convex set and that the restrictions of A1, A2 (i) and A3 to :~o are precisely the axioms of one of the versions of the von Neumann-Morgenstern axiom system. They thus imply [Theorem 1 in Fishburn (1982, Chapter 2), and in Ferguson (1967, Chap- ter 1)]:

PROPOSITION 1: There ex&ts a linear utifity, Vo, on (JJco , ~); Vo/s unique up to a positive affine transformation.

3.2. Expected Utility Representation for Preferences in ~3co

3.2.1. In EU theory, the expected utility representation U(P) =Epu results straightforwardly from the linearity of U and expression P = ~ ~_ ~ Pi Ici, once von Neumann-Morgenstern (vNM) utility u has been defined by u(c) = U(Ic).

In our model the expected utility representation of preferences in ~co has to be achieved differently since ~E/co does not contain degenerate distribution Ic unless c-Co; two-outcome distributions (1-~t) Icp+/~I~


however do belong to ~ o for c p - Co, c ~ Co and 0 < # < 1, and we shall take advantage of this fact to prove:

3.2.2. PROPOSITION 2: There exists a function, Uo, defined on [Co, --'], such that Uo, defined on +3co by

(1) Uo( P) = Epu o,

is a linear utility on ( ~ o , ~)" This property characterizes Uo up to a positive affine transformation, and implies that Uo is a utility on ([Co, ~ ] , ~).

Proof." By Proposition 1 there exists on ( ~ o , ~ ) a linear utility, Uo, satisfying Uo ( I J = O .

A "vNM utility-like" function, Uo, associated to Uo, can be defined on [c o, ~ ] by

1 (2) Uo ( c ) = - Uo ( (1 - i t ) Ico+itl~), for any 0 < i t < l ,

It

since this ratio is independent of It, which can be seen as follows: For

c ~ Co and 0< i t < i t ' < 1,

( 1 - i t ) Ico + i t lc=(l - it ) Ico +- ~ [ (1- i t ' ) I~o + U'Ic], It'

hence, since Uo is linear and Uo(Ico )= O,

Uo(( l - I t ) Ico + Itlc)= ~, Uo((1-It ' ) Ico + It'Ic).

Let us show that Uo is a utility function on ([Co, -->], ~) : For c', c" in [Co, ~ ] and 0 < I t < l , it results from A2(i) that c ' - c " implies (1 - I t ) I ~ o + # I e - ( 1 - # ) Ico +itlc,,, and from A4 that c' >-c" implies ( 1 - i t ) Ico+/~I c, >- (1 - I t ) /co +#I t " ; hence, c' ~ c" if and only if Uo((1 - / t ) /co + Itlc') i> Uo((1 - It) Ico + Itle,) or, equivalently, Uo(C') >1 Uo(C").

Consider then any P in .~co. P can be written in the form

P=polcp + ~ pilci ' i = 1

with Po > O, ~ P i : 1 - Po, ci ~ Co for i = 1 ..... n, and c p - Co, i = 1

C H O I C E U N D E R RISK A N D T H E S E C U R I T Y F A C T O R 177

hence also as

~-~ P~ ~, (polcp + ( 1 - Po) Ici), P = t=~ 1 - p o

i.e. as a c.l.c, of two-outcome lotteries (1 - /1) Ice+ltlci.

By A2(i), polcp+(1-Po)Ici-poIco+(1-Po)Ici , hence, by (2), Uo(polcp + (1 - Po) Ici) = (1 - 1)o) Uo(Ci). By using the expression of P and the linearity of Uo, Uo(P) = ~. i n- _ 1 PiUo(Ci), hence, since Uo(Cp) = Uo(Co) = O,

Uo(P) = E puo. Thus ( ~ o , ~ ) has an expected utility representation. Moreover, it is easily seen that Epwo = aUo(P)+[3, for all P in '~co, if and only if wo=aUo+[3 (for the "only if" part, give P values Ico and ( 1 - # ) Ico +l~Ic), hence that the expected utility representation of ( co, ~ ) is unique up to a positive linear transformation. Q.E.D.

3.3. Relations Between Expected Utility Representations o f Preferences in Different ~ ' s .

3.3.1. Close links between the expected utility representations of preferences in different ~ ' s are established by the following theorem.

T H E O R E M 1. There exist functions u and U, defined respectively on outcome set d a n d lottery set ~, and related by

U(P) = Epu, for all P in ~,

such that: (i) U is a linear utility for preferences among lotteries ensuring a common security level, i.e., when c p - c 0, then P ~ Q i f and only i f u (P) >>. U(Q);

(ii) u is a utility representing preferences under certainty. Moreover, u and U are unique up to a positive affine transformation.

Proof. (i) Let us first relate the expected utility representations of two sets of lotteries ~co and .~ , with c 1 ~ c o. Proposition 2 applies to "PCl as to ~co-

Let then Ul, defined on [cl, ---'], be such that Ul, defined on ~Jc~ by Ul(P)=EpUl, is a linear utility on (~C~cl , ~ ) .

Consider on the other hand u 0 of Proposition 2; since Epu o is defined for P in . ~ , (1) can be used to extend the definition of Uo to ~cl-


Let us show that we thus obtain a linear utility on ('~Cl, ~,~)- By A2 (i), given any P, Qin~c, and 0 < 2 ~< 1, P ~ Qi f and only if ( 1 - 2 )

/co + 2P ~ (1 - 2 ) Ico + 2Q; since these last lotteries belong to "~o, and Uo is a linear utility on ~ (1), Q if and only if ( co, ~)sat is fying P ~ ( 1 - 2 ) Uo( Co) + ~. E puo >f (1 - 2) Uo( Co) + 2 E puo, which, according to the definition of Uo on , ~ , is equivalent to Uo(P)>1 Uo(Q).

It results then from Proposition 2 that the restriction of Uo to ~ 1 and that of Uo to [cl, ~ ] are images of, respectively, U1 and Ux by a common positive affine transformation.

(ii) Let us now show how the preceding result generalizes to the whole

of 5J.. Consider outcomes Co and ca satisfying Co ~( c~ (if ~ is trivial on d~ the

conclusions below will remain clearly valid). It results from Proposition 2 that, for any c ~ Co, including Co itself,

(~c, ~ ) has an expected utility representation, and that this utility, Uc, can be compelled to satisfy uc(co)= ko and uc(cl)= k~, for given ko < kl.

It results moreover from part (i) that for c" -< c" ~ Co, ue, is identical to the restriction of Uc, to [c", ~ ] ; hence, in particular, that

u~,(c") = ue , (c") . Let us define a function u o n , b y : u(c)= u~(c), for c ~ Co; u(c)= Uco(C)

for c >- Co. It is straightforward that, for any c ~ c o, u's restriction to [c, -->] is

identical to Uc. Thus function U, defined on .~ by

U(P) =Epu,

is a linear utility on (~c, ~ ) for all c ~ Co. By part (i), this is also true

when c >- Co. Function u is obviously a utility on (~, ~). Moreover, it is unique up

to a positive affine transformation (provided the values of ko and kl are freely chosen). Q.E.D.

3.3.2. Theorem 1, which sums up all the conclusions we have so far obtained, gives no indications concerfiing the comparison of lotteries which offer different security levels. We shall be able to derive stronger conclusions in Section 4 by using the full force of our axiom system, in particular implications of independence axiom A2 (ii) which has not so far been used.

C H O I C E U N D E R RISK A N D T H E S E C U R I T Y F A C T O R 179

4. G E N E R A L R E P R E S E N T A T I O N OF P R E F E R E N C E S

4.1. Introducing Examples

The main features of the representation theorem characterizing preferences compatible with the axiom system can be anticipated through the examination of a few examples.

In the following, ( = R +, pp is the mathematical expectation of lottery p, /> L is the lexicographic ordering of ~ 2 , and [c] denotes the integer part of real number c. Five different weak orders are respectively defined by:

P ~1 Q if and only if P ~2 Q if and only if P ~,~3 Q if and only if P ~4 Q if and only if P ~ 5 Q if and only if

(Cp+~p) Cp~(CQ+~Q) co; (r ~ p) ~p ~ (cQ"b l..IQ) ~Q; 2 g p - cp~>2#Q- co; (Cp, #p)>~L(c o, ~tQ); ([Cp], Cp "at ].lp) ~ L ([CQ], Co "at l.l Q);

It is straightforward that axioms A1, A2 (i) and A3 are satisfied in the five cases; moreover functions u and U which have the properties of Theorem 1 are easily found; they are: u(c)=c for all c in •+, and U(P) =#p for all P in ,~

Let us now turn to independence axiom A2 (ii). It is clear that ~1, which is defined by utility VI(P ) = (Cp + IIp)Cp satisfies A2 (ii); ~2, defined by Vz(P ) = (Cp + pp)gp, however, does not satisfy this axiom (take: P' and Q' such that c~=cQ,=O; ~tp=x/2; #Q,= x/6; P" and Q" such that cp, = CQ,, = 1; #p,, = 1; ~t Q,, -= 2 and ~, = 1/2); it is in fact easily checked that A2 (ii) requires that utility V, when it exists, have the form

(3) V(P) = a(cp)Epu + b(cp).

Dominance axiom A4 however adds some further limitations by putting restrictions on functions a and b: ~3, which is defined by a utility of the form (3), V3(P)=212p-Cp, does not satisfy A4 (take P=I1 and Q = V3 I0 + 2/3 I0; the required restrictions will be derived in Subsection 4.2.4.

Thus it appears that preference orderings defined by utilities of the form (3) - with some restrictions - will satisfy the axiom system. The converse is however not true: ~4, which complies with all the axioms has no utility representation whatsoever (this well-known property of the lexicographic ordering of ~2 obviously remains valid for ~4).


In limiting cases of preferences, such a s ~,~4, a lower security level is never counterbalanced by a higher expected utility. Tradeoffs between security level and expected utility are however allowed by the axiom system and may exist to variable extents; ~5 exhibits tradeoffs of this type and is such that in subsets of J in which they occur, a utility representation of the form (3), Vs(P ) =Cp+ Up, exists. This property shall prove to be general.

4.2. Representation Theorem for Overlapping Preferences

4.2.1. Let us first restrict our attention to sets of lotteries where any difference in security level can be compensated for.

Formally, given outcomes Co and Cl such that Co -< Cl, preferences in ~co and.~cl are said to overlap if there exists Po in ~ o such that Po ~ 1~. Given a subset C of~, let J c = I P : P e ~ c for some c in C]; preferences in .~c are said to overlap when, for every Co, Cl in C such that Co ~( Cl, preferences in ~ o and ~S*~ overlap.

4.2.2. We need now establish two technical lemmas; the first one states a general property which will be used in several proofs; the second one concerns overlapping preference sets specifically.

LEMMA 1. I f P, Q and R belong to ~, c p - cQ, and P >- R >- Q there exists a unique ~ such that 0<~. < 1 and ZP+ (1 -~,) Q - R .

This lemma follows from A1, A2(i) and A3; proofs of the similar property in EU theory ((e.g., Ferguson (1967, p. 15); Fishburn (1982, p. 18)) can be readily transposed.

We can then prove:

LEMMA 2. Let Co and cl o f E be such that Co -< Cl and suppose that preferences in ~co and ~Cl overlap; then, for some Pco in :~co, Pco >" Icl, and, for every c in [c o, Cl], there exist Pc and Qc in ~c such that P c - Pco and Qc-Icl; moreover, preferences in ~f2,[co, cd overlap.

Proof. Let Rco in ~co be such that Rco ~ Icl. By A4, there must exist c2 >- cl such that Rco =alco + [jlc2 + (1 - a - [j) Q, with a > 0 and [j > 0. Let Pco = ( a - e ) Ico+ (fl +e) Ic2+ (1 - a - [ J ) Q, with 0 < e < a ; Pco >.O Rco, hence by A4 again and by AI , Pco >" Ic~.

Let c >- Co. Let us write Pr = F. ki= 1 ),i Iei + F. 'I= ~ + 1 )~i Ic i, with c i -< c if and only if i ~< k; necessarily, ~/k= 1 ~-i> 0; hence Sc defined by Sc = ( ~. ~= 1


hi) I c "at- ~ n= k + 1 hi Ic i belongs to .~c and S c ~.o Pco, hence by A4, S c >- Pco. Let moreover Sco be defined by Sco = Pco.

For Co -( c ~ cl, Sc >- "~ >" Ic; thus, by Lemma 1, there exists Pc = aSc + (1 - a) Ic such that P c - Pco. Similarly, for Co ~ c -< Cl, Sc >- Icl ~-I c implies the existence of Qc=~tSc+(1-#) Ic such that Qc-Ic~. Obviously, Pc and Qc belong to ~ . Let moreover Qc~ = Ic~; then, the first part of the lemma is completely proved.

The second part follows readily, since, for any c', c" in [Co, cl], Pc' >- Ic~ and Ic, ~ Ie,. Q.E.D.

4.2.3. By using for the first time independence assumption A2(ii), we shall now be able to prove the existence of a representation of preferences in ~tco. c~l by a utility of the form suggested by the examples of Section 4.1.

PROPOSITION 3. Suppose that preferences in .~9co and Jc~ overlap. There exists then a utility u on ([Co, ~] , ~) and a utility V on ( ~Jtco, cd, ~ ) related by

(3) V(P)=a(cp)Epu+b(cp), for all P in ~tco, cd,

where a(Cp) and b(cp) only depend on ce's indifference-class, and a(cp) > O.

Moreover, u and V are uniquely determined up to independent positive affine transformations, and to transformations u' =,~u + lt, V' = a V+ [3 correspond transformations a' = (a/;Oa and b' = ab - (~a/)~) a + 13 on a and b.

Proof." (i) Let u be any utility on ~ ~) with properties of Theorem 1, and let, for every c in [Co, cl], lotteries Pc and Qc have properties of Lemma 2.

For every c in [c o, cl], Epcu>EQcU, hence there is a unique solution a(c), b(c) to system

(4) a(c)Ep u + b(c) = 1, a(c)Ea u + b(c) = O,

and a(c)>0; moreover c ' - c " implies Epeu=Epe,U and EQeU=EQc,,u, hence a(c')= a(c") and b(c')= b(c").

Relation (3) defines then a function V on "~co, c~l which satisfies V(Pc)= 1 and V(Qc)=0 for every c in [Co, Cl].


By Theorem 1, V i s a linear utility on each (~c, ~) ; thus, for R in.Yc, V(R) =~,, with: 2 < 0 and Qc- (1 / (1 -~ . ) ) R + ( - M ( I - ~ . ) ) Pc, if R-< Qc; 0 ~< ~. ~< 1 and R - (1 - ~) Qc + 2P~ if Qc ~ R ~ Pc; ~. > 1 and Pc - ((~- - 1)/~) Qc+ (1/).) R if R >-Pc.

We have to show that V i sa utility on (~co, cd, ~) , i.e., that, for any c', c" in [Co, c1], R' in ?J~e and R" in ~Jc", R' ~ R" if and only if ~ '>~" , with ~' = V(R') and ~," = V(R"). The proof is straightforward when R', Qc" and Pc' are ordered differently than R', Qc',, Pc"; the proofs in the remaining cases are similar, and we shall only examine the case where R' ~" Pc' and R" >- Pc", hence where:

~ ' - 1 1 Z " - 1 1 - - - - - R " . (5) Z' QC'+-~R'-PC'-PC', Z" Qr

Since Qc'-Qc' , , it follows from A2(ii) that R ' ~ R" if and only if ( (~"- 1)/~.") Qe + (1/~.") R' ~ ((~."- 1)/~.") Qc,' + (1/~.") R", which, by (5), is equivalent to ( 0 . " - 1)/M') Qc" + (1/~") R' ~ ( 0 . ' - 1)/~.') Qe + (1/~.') R', and also, since R' >- Qc', to (1/;t") ~>(1/~.'), i.e., to ;t'~>A.".

(ii) If V' is another utility on ('~co, c11, ~ ) satisfying (3) for some functions u, a' and b', then, by Theorem 1, which applies to (?~o, cq, ~) , u'=s for some ; t>0 and 12; moreover V' is linear on each .~ , hence completely determined by its values at Pc and Qc; since these values do not depend on c, we can put 1/' (P~)=a +/3, V'(Q~) =/3, with a > 0 , thus

a'(c)Epc (~,u + 12) + b'(c) = a +/3, a'(C)EQc (A,u +12) + b'(c) =13;

comparison with (4) yields

a b' = 12a a ' = - a , a b - - - a + / 3 and V'=aV+fl.

Conversely, it is straightforward that for arbitrary ;t > O, a >0, 12 and/3, this transformation preserves the properties of u and V. Q.E.D.

4.2.4. Function Vof Proposition 3 clearly satisfies the restrictions of A1, A2 and A3 to (~ko, cd, ~ ) but not necessarily that of dominance axiom A4 (see examples in Section 4.1). The following proposition determines the restrictions imposed to V by the dominance requirement:


PROPOSITION 4. Let u, a, and b satisfy the properties stated in

Proposi t ion 3 and let V be def ined in ~tco, ~1] by (3). The fo l lowing statements are then equivalent: (I) I f P ' and P " belong t o . ~ o , ~d and P ' ~ .o p, , , then V(P') > V(P").

(II) I f c' and c" belong to [c o, Cl] and c' >- c", then

(6) b(c") - b(c') <<. Inf (a(c') - a(c"))u(c). C ~ C ~

Proof . (i) Let us first note that, since u is a utility on ([Co, --'], ~) , relation (6) is equivalent to relations

b(c") - b(c') <~ (a(c') - a(c")) u(c') if a(c') >1 a(c"), and

b(c") - b(c') <~ (a(c') - a(c")) ~ if a(c') < a(c"),

where u=Supc u(e) is an element of fRw[+ oo], hence, that (6) is also equivalent to

(7) b(c") - b(r <<. Inf (a(c') - a(c"))y. u(c') <~y <~

(ii) Suppose that (II) holds. If P ' , P" belong to ~co, cd and P ' )..o p,,, then cp, =c ' and OR,, =c" for some c' and c" such that Co ~ c " ~ c' ~ el; moreover E p , u > E p . u and u(c')<~ Ep,u<~ a.

It thus results from a(c")>0 and from relation (7) (which remains trivially valid when c ' - c " ) that

V ( P ' ) - V ( P " ) = a ( c ' ) E p , u - a(c") Ep,,U + b(c') - b(c")

> ( a ( c ' ) - a(c")) E p , u + b(c') - b(c") >>. 0;

thus (II) implies (I).

(iii) To prove the converse, let us now suppose that (I) holds.

Given c' and c" in [c o, cl] such that c' ~ c", u' in [u(c'), u~, and u" in [u(c"), u~ such that u' > u", there exist P ' in . ~ and P " in -~e' such that E p , u = u ' , E p . u = u " , and P ' ~ . 9 p , , . (It is easily checked that P ' =,~,I e + (1 -~.) I e and P " = ~Ie, + vie + (1 - II - v) I c have the required properties for ~ such that u(c~ > u' and appropriate values of ;t,/~ and v satisfying ~ + v~>;t). Thus, according to (I),


a(c') u' - a(c") u" + b(c') - b(c") = V(P') - V(P") > O.

Since this must be true for all values of u' and u" which can be associated with c' and c", we have then

b(c")-b(c')<,inf [a(c') u' -a(c") u": u(c")<~u"<u' and u(c') <~ u'l,

from which relation (6) easily follows by using the fact that

a(c')u'- a(c") u"= (a(c')- a(c")) u' + a(c") (u'- u") and a(c") > 0. Q.E.D.

The following property results straightforwardly from relation (7):

COROLLARY 1. Let u and V satisfy the properties stated in Propo- sition 3. Then, at a given expected value o f u, V is a non-decreasing function o f the security level, i.e.,

Ep,u=Ep,,U and Cp, ~ cp,, imply V(P')>~ V(P").

REMARK 1. Preferences in ~co and ~3c~ defined by V satisfying (3) overlap if and only if

(8) a(co) ~+ b(co) > a(ca) U(Cl) + b(cO.

This inequality states namely that, for P in ~co, Sup V(P)> V(I~I ). Unlike (6), condition (8) is not a restriction imposed by the axiom

system; as we shall see there indeed exist utility representations of the form (3) on subsets ~ c of ~ i n which preferences do not overlap.

4.3. Representation Theorem for Connected Preferences

4.3.1. Consider the following example: preference ordering ,~6 is characterized by utility function V 6 defined by

2 V6(P) = cp + Epu where u(c) = - t a n - l (c), c in ~ = R +,

7[

and t an - 1 denotes the antitangent function. The exact bounds of I/6 for P in ~c are thus c+u(c)<<.V6(P)<c+l; hence (see Remark 1), for Co < cl < c2 such that

C H O I C E U N D E R RISK AND THE S E C U R I T Y F A C T O R 185

C 1 + U ( C l ) < C 0 + 1 ~C2+U(C2)<(Cl + 1,

preferences overlap in ~r and ~ l as well as in : ~ and ~ 2 but not in Jco and JJc2; yet a utility function of the form (3) - II6 is itself - exists

not only on ~co, Cll and ~tc~, czl but also on ~ttco, czl. We shall see that this property is in fact general: the junction of utility

functions of the form (3) which exist on adjoining subsets o f ~ c a f i always be realized (by merely performing an affine transformation on one of the functions).

First, let us define a natural extension of the concept of overlapping preferences as follows:

Let Co and c belong to g a n d Co -< c. We will say that preferences in ~co

and ~c are connected when there exists a connecting sequence (ci), i = 1,

.... n, between Co and c, i.e., outcomes ci such that Cn-C and that for every i = 1 . . . . . n, c i_ 1 "~ ci and ~ i - ~ and ~ci overlap. More generally, given a subset C of g, we will say that preferences in ~ c are connected

when, for every c o -< c 1 in C, preferences in ~co and ~c~ are connected. We must also introducethe maximal subsets of ~ i n which preferences

are connected: for every c in -f, let G(c) = [c' E~ : d - c, or ~ and ~ e are connected, or ~e and ~c are connected] and J = [G ~ 2 ~" : G = G(c) for some c i n g ] . Whenever G', G" belong to Wand P '> - P " for all P ' in J o ' and P " in ~o", we will write ~c ' >" ~a" , and thereby define relation >-.

LEMMA 3. J is a parti t ion o f ~, and, f o r every G o f J preferences in

.Ya are connected; moreover relation ~= (i.e., ~- or = ) is a linear order on

Proof . By Lemma 2, if some ~co and.~c~ overlap and c o -< c 2 -~ Cl, then Jco and ~c2, as well as .~c2 and .~c~, also overlap. By using this property to construct the required connecting sequences as modifications of given ones, it is easily shown that, for any c' ~( c ~( c", ~.~ and.~c,, are connected if and only if ~d/c' and -~c on the one hand, ~c and ~c- on the other hand, are themselves connected.

This implies that if G' and G" belong to J and have a non-empty intersection, then for any c ' in G ' and any c" in G" such that c" >- c',

.~, and Jc,, are connected, and vice versa. The first part of the lemma follows immediately.


For the second part, consider G' and G" of J,, with G' distinct from G"; let e' belong to G' and c" to G" and suppose that ~'~( e". It follows from the above: first, that for any c" in G",e ' -< c"; second, that for any P ' in ~6,, and any Ic,, in 3c,,, P ' ~( Ie,; lastly, that for any P ' in J c ' , and any P " in ~c", P ' ~( P". The transitivity of ~ being obvious, the second part of the lemma follows immediately. Q.E.D.

In some of the examples already considered, J can be easily determined: J = / ~ ] for ~1; J=/[c] ; cc-~l for ~4; and J = [ [ n , n + l ) : n ~ N ]

for ~s-

4.3.2. We shall now prove that the representation theorem for overlapping preferences extends to connected preferences as follows:

PROPOSITION 5. Let G be any member o f , J and let ~ be the corresponding connected preference subset o f ~ ; let moreover G* = Ic ~ : c ~ c' f o r some'c' in G].

There exists then a utility u on (G*, ~ ) and a utility V on (,~G, ~ ) , related by

(9) V(P) = a(Cp) Epu + b(cp) f o r all P in ~G;

moreover functions u, V, a and b possess all the properties o f the corresponding functions stated in Proposition 3.

Proof. (i) If (~ ~ ) has a greatest element ~, there exists G in ,•such that G = [c e ~ : c - ~; ~ has only one indifference class, so that the proposition holds trivially for G. We may thus assume that G contains no greatest element of (~, ~) , which implies that for any Co in G there

exists Pco in ~ o such that Pco ~" I~o. Let u be any utility on (~, ~ ) satisfying Theorem 1. (ii) Let us define a function V o n - 9 c as follows:

( co, ~ ) Let V coincide in :~3co with the unique linear utility, V o, on which satisfies Vo(Ir 0 and Vo(Pco )= 1.

For P in ~c and c >- c o, let us introduce a connecting sequence (ci), i = 1, .... n, between c o and c. By Proposition 3, a utility II,- satisfying (3) for the chosen u exists on each ( , ~ _ ~, ~1, ~) , and the Vi's are successively completely determined by the requirement that Vi and V i_ ~ coincide in

( i= 1 ..... n). Let us then set V(P) = Vn(P). " C i - - I

C H O I C E U N D E R RISK AND THE S E C U R I T Y FA CT O R 187

We have to show that V(P) does not depend on the particular connecting sequence used. Note first that if ~ci_ 1 and ~c,-+ 1 overlap, the unique utility Von ( ~ c i - ~, ci+ 11, ~ ) which satisfies (3) and coincides with V i_ l and Vi in .?~i_ 1, coincides, according to the unicity property of Propo- sition 3, with Vi in ~ i - 1 , ~iJ, hence with Vi+ 1 in ~,. , and therefore with Vi + 1 in.~c~, c;+ 11" This implies that, more generally, any sequence which is finer than (ci), i.e., contains in addition some intermediate outcomes, leads to the same value for V(P). Since there always exists a connecting sequence which is finer than each of two given distinct sequences, these necessarily also lead to the same value for V(P).

For P in :~/and c -< c o, V(P) is defined similarly by using a connecting sequence between c and Co.

(iii) Let us show that, in .~o, P ' ~ P " if and only if V(P') >1 V(P"). Since Visa utility on every,~c, this is true when Cp,- Cp,,. We may thus assume t h a t Cp, -~ Cp,,.

There is then a connecting sequence (ci) i= 1 ..... n, between Cp, and Cp.. By the same argument as in (ii), if JJcp, and r overlap, the unique utility on ( JJ[cp,, cp,,l ~) , which satisfies (3) and coincides with Vin yS~cp,, coincides with V everywhere in .~p , , ~p,,], and V has thus the required property.

If, on the contrary, ~Jcp, and ~Jcp,, do not overlap, then Ice, ~ P ' -< Ice,, P" , hence P ' -<: P" . More precisely, for some j <. n, Ic;_l ~ P' "< Icj

~Icp,, and therefore, since V is a utility on every (JIr ci], ~) , V(P') < V(Ic) <~ V(Icp. ) ~ V(P"), hence V(P') < V(P"). V is thus a utility on ( J ~ , ~) ; its other properties and the unicity property of u's restriction to G* result directly from Proposition 3. Q.E.D.

REMARK 3. Proposition 4 implies that if c', c" belong to G, c' ~( c", and Jc ' and ~ , , overlap, condition (6) must be satisfied. Note that A4 entails no restriction when ~c, and ~ , , do not overlap; in fact, (6) is automatically satisfied in that case, since (see (8))

a(c") ~ + b(c") <~ a(c') u(c') + b(c').

4.4. General Representation Theorem

4.4.1. The conclusions of Proposition 5 apply to preferences in -~c for


every G in ~4"; bringing them together with the conclusions of Theorem 1 and Lemma 3, which concern preferences in the whole of ~ we obtain a general representation theorem which characterizes preferences satisfying our axiom system.

The caracteristic property stated in this theorem involves a partition of K'which could be either ~ o r any partition which is less fine than It will be clear that the theorem remains valid with ~ r e p l a c e d by ~ It may however be easier to check that some -~, rather than . J itself, has the required properties.

For an example where the properties of Proposition 5 extend beyond sets G of ~G suppose that ( = N+ and define ~ 7 in J b y utility function I"7 such that

2 Vv(P) = [cp] + - E p t a n - t.

7~

Since preferences in.~e and ,~e' only overlap when [c'] = [c"], J = [[n, n + 1] : n e H]; yet V 7 is defined, and of the form (9) in .~ i.e., in ~ H with H = ~ ; thus, in Theorem 2 below, characteristic property (II) is satisfied not only for -)F~ = .~ but also for W.=-- [C].

4.4.2. Preferences complying with the model will thus be characterized as follows:

T H E O R E M 2. Given a preference relation ~ in .~ set o f aU lotteries on an outcome set G the following two statements are equivalent:

(I) (J~, ~ ) satisfies assumptions AI, A2, A3 and A4. (II) There exist real functions u, a and b on ~ and a linear order, >-,

on a partition ~ o f G such that: (i) for all P, Q in ;4

(10) P ~ Q if and only i f H(cp) > H(ca) or [H(cp)=H(c o) and V(P)>~ V(Q)],

where: cp is a u-minimal outcome of P, i.e., P(Icp])>0 and P ( [ c e r : u(c) >~U(Cp)])= 1; H(cp) is that member of ~s contains co; and V( P) = a( c p)E pu + b( c p) and a( c p) > O.


(ii) the fo l low&g consistency relations hold: fo r any c', c" in u(c') >>. u(c") implies that H(c') ~= H(c"); u(c') = u(c") implies that a(c') = a(c") and b(c') = b(c"); H(c') =H(c") and u(c')> u(c") imply that

(11) b(c") - b(c') ~ In fu(c)/> u(c') (a(c') - a(c"))u(c).

Moreover, the properties in (II) imply that u is a utility on (~, ~ ) and is unique up to a positive affine transformation; they also imply that, i f I ~ : G e J I is the partition o f ~ i n t o connected sets, .:r at least as f ine as ~ , and that the restriction o f V to each 3 c is unique up to a positive affine transformation.

Comment: Relation (10) expresses the partially lexicographical character of preferences in the model: P is strictly preferred to Q whenever the security level Cp it ensures belongs to a higher class of security levels in ,W'then Co; on the contrary, when Cp and c o belong to the same class in ~.,' other characteristics of P and Q are also taken into account, so that the scale may tip in favor of the lower security level lottery.

Note that V(P) does depend on the probability P([c ~ : u(c)= U(Cp)~) of obtaining the minimal outcome of P through the term Epu.

Proof. It results from Theorem 1, Lemma 3 and Proposition 5 that (I) implies (II) with o,Y=~

Conversely, assuming (II) to hold, it is easily seen that ~ defined by (10) satisfies A1, A2 and A3; moreover (11) enables the inference - by Proposition 4 - that A4 also holds, provided function u is a utility on (~, ~ ) , which fact can be proved as follows: first, u(c')= u(c") implies that V(Ic,) = V(Ic,,), hence that I e, -Ie,, and c ' - c " ; second, u(c')> u(c") obviously implies that I c, >- It,,, hence that c' >- c" when H(c') >- H(c"), but also when H(c' )=H(c") since in that case, by (11), a(c') u(c')+b(c')>a(c") u(c") + b(c"), i.e., V(Ic, ) > V(Ic,,).

Concerning the relations between dY~and J i n (II), it suffices to note that if ~Jco and ~cl overlap - and thus U(Co)<U(Ct) and H( co) <( H( c l ) - necessarily H(co) =H(cO. Therefore, if ;+~co and ~el are connected (i.e., c o and cl belong to the same member of W), all outcomes belonging to the connecting sequence, including c o and cl, must belong to the same member of ~ ' ; hence, J i s at least as fine as ~ .


Finally, the unicity properties of u and V follow from Theorem 1 and Proposition 5, respectively. Q.E.D.

5. OTHER PROPERTIES OF THE MODEL

5.2. Particular Forms and Variants of the Model

5.1.1. As already noted in section 2 when presenting the axiom system, our model generalizes EU theory, which corresponds to the particular case where there exists on (~, ~ ) a utility V of the form (9) with a and b constant functions; thus EU theory is the limiting case of the model where the security factor does not affect preferences at all.

Conversely, the main advantage of the model over EU theory is its ability to express a possible influence of the security factor. The most general form of the model may not however necessarily be required to do so.

For example, a decision maker who tries above all to limit the amount of his losses could have preferences of the form

P ~ Q if and only if Inf[ce, 0] > Inf[cQ, 0] or [Inf[cp, 0]=Inf[cQ, 0] and Eeu>~EQU];

alternatively, they could be of the form

P ~ Q if and only if Ice >>. 0 and CQ < 0] or [(cp i> 0 = CQ or coc o > 0) and Epu >1 EQU],

if he is chiefly anxious not to incur any losses - whatever the amount of loss involved.

In both these examples, where Sequa l s respectively [~ + ] u [[c] : c < 0] and [[R+, ~*_], preferences comply with EU theory in every connected preference set ~'~o, and functions a and b of (9) are constant functions.

We shall now determine the axiomatic requirements which characterize a submodel of this type and discuss their psychological plausibility.

5.1.2. Let us first consider the following independence assumption: (A2) (iii) I f P, Q, R and S belong to ~ ~ f [CR, CS] ~ S~p[ce, CQ],

and 0 <<. ). <~ l, then


(12) ~ P + (1 -&) R ~ ; tQ+ (1 -,~) R i f and only i f ,~P+ (1 - ~ ) S ~ Z Q + ( 1 - Z ) S.

We have:

PROPOSITION 7. Function a o f Theorem 2 is constant in every set G o f partition S i f and only i f preference relation ~ satisfies A2 (iii).

Proof. With P, Q, R and S as in A2 (iii), and ~ > 0, let Cp, CQ belong to the same G o f , s Functions u, a, and V of Theorem 2 are such that

V(XP+ (1 - A,)S) = V(~,P+ (1 -X)R) + a(cp)(1 -A,)[Esu -ERul , V(ZQ + (1 - Z)S) = v(~,Q + (1 - A)R) + a(co)(1 - A,)[Esu - ERu].

Thus: (i) If a(cp)=a(cQ), then V ( ~ P + ( 1 - 2 ) S ) - V ( Z Q + ( 1 - ~ , ) S ) = V(ZP + (1 - ~) R) - V(2 Q + (1 - ~) R), hence, since V is a utility on (.fiG, ~ ) , (12) must hold.

Since (12) obviously also holds when ~ = 0 or when Cp and CQ belong to different G's, the necessity of A2 (iii) is proved.

(ii) Suppose now that a(cp) ~ a(cQ) and that.~ce and.~ overlap, which, on account of A4, requires the existence of c in d r such that c o -< Cp -,( gand a such that 0 < a < 1 and aIcQ+ (1 - a ) I~e~(I~p-<aIco+(1 - a ) Ie; moreover, there exists, by Lemma l , /3 such that 0< /3< 1 - a and

I~,~ - c t I ~ +/3I~,~ + (1 - a - / 3 ) I~.

Therefore, for P = R = l c p , Q = ( a / ( 1 - / 3 ) ) I c o + ( 1 - a - / 3 / ( 1 - / 3 ) ) I o S = Ie, and ~ = 1 - /3, we get

V(A,P+(1-A)R)= V( ; tQ+(1-Z)R) and V ( Z P + ( 1 - A ) S ) = I/(2, Q + (1 - ~,)S) + [a(cp) - a(cQ)] (1 - Z) [u(~) - U(Cp)],

hence,

V(ZP + (1 - 2,)S) :/: V(ZQ + (1 - 2.)S).

Thus (12) cannot hold, unless a is constant in every [c', c"] such that


preferences in .~r c"] overlap; but in that case a is in fact constant in every G of .~ Q.E.D.

Since independent arbitrary positive affine transformations can be performed on V's restrictions to all G of J , the following corollary is straightforward.

COROLLARY 2. I f preference relation ~ satisfies A2 (iii), it can be required, in Theorem 2, that function a be any positive constant function.

5.1.3. The independence requirement can be further reinforced with: (A2) (iv) I f P, Q and R belong to .Z cR ~ ~ p [Cp, CQ] and 0 < Z <. 1,

then

(13) P ~ Q i f and only i f ~.P+(1 -A)R ~ AQ+(1 -~.)R.

We shall now prove:

PROPOSITION 8. I f preference relation ~ satisfies A2 (iv), then preferences comply with EU theory in/~Hfor every member H o f partition -~" o f Theorem 2.

Proof. Since A2 (iv) implies A2 (iii), we can use Corollary 2 and assume that a(c)= 1 for all c in

Consider then c' -,( c" such that preferences in ~Jtc', e'l overlap. There exists then P in-~c' and Q in .~Ye' such that P - Q , in which case, by (13), for every Z in (0, 1], Q - A P + ( 1 - ~ . ) Q; thus, u and Vof Theorem 2 are such that

EQu + b(c") = V(Q) = V()LP + (1 - ~,)Q) = E U A p + ( 1 _ z ) O + b(c') = ~,Epu + (1 - ,~) EQU + b(c'),

hence that b(c")-b(c')=;t[Epu-Eou], for all ), in (0, 1], which implies that b(c")=b(c') (and EpU=EQU).

Thus function b is constant in [c', c"], hence in every G o f , s and can be assumed to be constant in all d, for example, to be the null function. Since then V(P) =Epu for every P in .~ V is a linear utility in (.~/_/, ~ ) for every H of ~ . Q.E.D.

Since it is straightforward that the converse of Proposition 8 is also true, a characterization theorem (similar to Theorem 2) could also be stated.


5.1.4. To form an opinion concerning the reasonableness of independence assumptions A2 (iii) and A2 (iv), let us again consider a two-stage process of realization of the lotteries involved (see Figure 3). It turns out that the argument which can then b~ put forward to justify A2 (iii) relies on the validity of A2 (iv). Thus although A2 (iv) is more demanding than A2 (iii) from the logical point of view, they are equivalent from the psychological point of view: Both require that the decision maker not care whether Nature's choice of E or E c takes place before or after his own choice as long as: (i) the realization of E does not change the security levels involved; and (ii) conditionally to E ~ the choice made is immaterial. Thus assumptions A2 (iii) and A2 (iv) can be justified in a way quite similar to that used to justify A2 (ii), and seem about as plausible as this last axiom.

On the other hand, since it cannot be excluded that the submodel of Proposition 8 could remain partially subject to the descriptive and prescriptive defects of EU theory, one must turn to experimentation in order to gauge the advisableness of including A2 (iii) or A2 (iv) in the axiom

Prob(E)=,t Prob (E ~) = 1 -Z

~ P E P

Q k2(iv) ~ R

choLees),~ I A2CLLs

Q A2 Ci.v) ~ ~

[ ~c~,~l~i .... i 1 t

I same pair of security levels I at Nf and N~ for (8', 6"); at N z for (d', a"); at ~/2 for (~, ~')

Fig. 3. Justification of assumptions A2 (iii) and A2 (iv).


system. These assumptions can be tested indirectly by constructing functions a and b testing their constancy.

5.1.5. Gilboa (1986) has studied the properties of a more general model which reduces its independence requirements to axiom A2 (i), and proved a representation theorem which is quite similar to Theorem 2, the main difference being that function V is of the general form

V(P) =fH(EpU, U(Cp)) in each 3 H,

where fH is continuously increasing (resp. non-decreasing) with respect to its first (resp. second) argument. Gilboa also proves that the adjunction of A2 (iv) to his axiom system is necessary and sufficient to obtain the special form V(P)=Epu, which shows that in our axiom system the adjunction of A2 (iv) makes A2 (ii) redundant.

5.2. An Explanation o f Risk Aversion by the Certainty Effect

In this section we assume that (~ ~ ) = (JR, >=). According to the standard definition, the decision maker is risk averse (resp. risk neutral) when, for every P in ~, I~p ~ P (resp. I~p-P) .

Let us moreover consider a weaker concept of neutrality: we shall say that the decision maker is risk neutral at a given security level whenever,

(14) for every c in C, for all P, Q in ~c, P ~ Q if and only if I .p ~ I . O"

Since Iup ~ IuQ amounts here to I~p>>.UQ, i.e., to Epu >>.Eou, where J is the identity mapping, risk neutrality at given security level can be characterized by the property that function u in Theorem 2 is the identity mapping: u = J .

It turns out that:

PROPOSITION 6. A decision maker who is risk neutral at a given security level is necessarily a risk averter.

Proof. For any P in ~, cp <~ ltt,. If cp = I~p, then, by (14), Icp- P, hence I~p-P .

If, on the contrary, cp<#p, i.e., c p C #p: either H(gp)~-H(ce), in which case, I~p >-P; or, H(gp)=H(cp).

C H O I C E U N D E R RISK AND T H E S E C U R I T Y F A C T O R 195

In this last case, -'~co and ~ue necessarily overlap, since ~= + ~ ; thus Corollary 1 applies with P'=Iup and P"=P; hence V(Iup)~> V(P)and I~,p~P. Q.E.D.

This result calls for the following comments: In EU theory, it is generally thought that the shape of the von Neu-

mann-Morgenstern utility, which characterizes risk attitude, results, on the one hand, from the shape of a cardinal utility under certainty (which possibly expresses the diminishing marginal utility of money) and, on the other, from risk attitude proper. Yet the model neither distinguishes between these two components, nor does it allow to link up pure risk attitude with known psychological phenomena.

In Yaari's (1987) dual theory, however, it is explicitely assumed that the cardinal utility is linear (constant marginal utility of money), and risk attitude is entirely explained by probability distortion: risk aversion becomes characterizable by the convexity of the weighting function which distorts cumulative probabilities.

Proposition 6 shows that there exists another possibility- offered by our model- that of explaining risk aversion entirely through the security factor, a

REMARK 3. The argument of the proof of Proposition 6 remains valid if I~p is replaced by any lottery P ' such that ltp,=ltp and Cp,>~cp. Thus decision makers who are risk neutral at a given security level never consider as favourable mean-preserving "spreads" of their probability distributions on the outcome set. 3

5.3. Descriptive Limitations of the Model

Although our model is primarily a normative model, its defects as a descriptive model have to be detected before establishing assessment procedures for its prescriptive use.

Let us first note that some paradoxes of EU theory necessarily remain paradoxical with respect to our model, namely: (i) behavior which is inconsistent with the existence of a preference ordering, such as in Problems 11-12 of Kahneman and Tversky (1979) and Experiment 5 of Hershey, Kunreuther and Schoemaker (1982); (ii)behavior which is inconsistent with weak independence requirements, such as Problems 7-8


of Kahneman and Tversky, i.e. examples of Common Ratio Effect where all lotteries involved offer the same security level.

On the contrary, other examples of the Common Ratio Effect, such as Problems 3-4 of Kahneman and Tversky, or the famous Allais (1979)

Paradox, which both involve lotteries offering different security levels, display behavior which is no longer inconsistent with our model; in fact,

conditions on functions a, b and u such that the phenomenon necessarily takes place are easily determined.

Thus, it appears that our model indeed accounts for those deviations

from EU theory which are generally ascribed to the security factor (which is the very reason it was developed) but not for deviations such as those first mentioned, which must be attributed to other psychological factors,

namely the framing effect and oversensibility to small probabilities.

5.4. Prescript ive Use o f the M o d e l

In this section we assume that (~,, ~ ) = ( R , />).

5.4.1. As se s smen t s

Let us first remark that relation

V(P) = a(cp)Epu + b(cp)

can be set in the form

(15) V(P) = a (cp )EAu - u(cp)) + v(cp),

if function v is defined on ( b y

(16) V(Cp) = V(Icp ) =a(cp)u(cp) + b(cp).

Thus the knowledge of a, u and v is equivalent to that of a, u and b. Let us now see how a, u and v can be constructed in an interval [Co, ~].

Since u is a von Neumann-Morgenstern utility in (-~co, ~) , it can be assessed by any method which only involves the comparison of lotteries belonging to 3c o, e.g., by the lottery equivalent method of McCord and de Neufville (1986). The assessments of a and v however necessarily require the comparison of lotteries belonging to different ~ ' s .


For example, given c in (Co, c~, and fixed probabilities ff and f, one can

try to successively determine probabilities:

(i) p, such that p Ie+ (1-/9) I c o - ~ I t + ( 1 - P D / c o ; (ii) q, such that q I c + ( 1 - q ) Ico-Ic;

(iii) r, such that r l e + ( 1 - r ) 1 c - f i e + ( 1 - r 3 Ico;

Unlike p, q and r do not necessarily exist; in fact, if ~S~o and ~ do not

overlap, they definitely do not exist. If they exist for some c', they also exist for every c in [Co, c']. Functions a and v are then determined in [Co, c'] - and u in [Co, ~] - provided the degrees of freedom of the model

are taken into account by setting conditions

U(Co): O, u(~ = 1, V(Co) = 0, and a(co) = 1.

It results then from (15) that:

(17) u(c) = P ; v(c)=q; a(c)= r - v ( c ) p r(1 - u(c))"

Functions a and v can be extended to the whole of [Co, ~] by further

assessments of the same type.

REMARK 3. EU theory would expect (i) and (ii) to be alternative methods

for constructing the same function - the vNM utility; EU theory is in fact the submodel characterizable by: u(c)= v(c) and a(c)= 1 for all c.

5.4.2. Sequential Decision Making

Let us show briefly how an optimal strategy, i.e., optimal sequential

decisions, in a decision tree can be determined by working from the terminal nodes to the starting node. 4

Let a given node N have successors S k ( k = 1 ..... K) where optimal subsequent decisions respectively secure prospects described by lotteries Pk.

(i) If N is a decision node, one can choose at N the node S~. which is to succeed to N, but cannot enforce subsequent decisions at later stages which would not make up optimal substrategies at every node. Thus, the best prospects which can be secured at N correspond to lottery Pk*--S~up Pk. The determination of k* only requires, according to

A ,


representation Theorem 2 and relation (16), that four characteristics of Pk : Ep, u, a(c,o,), v(cp,), and H(cpk), be known for each k.

(ii) If N is a chance node, and Sk has probability tk of succeeding N, prospects at Nare described by lottery R = Z~ ~.k P, , the four characteristics of which can easily be deduced from those of the Pk's since

E R u = Z k ~kEpkU and CR=Inf k Cpk.

Thus sequential decision making with our model only requires: (i) at each terminal node, the evaluation of u(c), a(c), v(c) and the determination of H(c) for the final outcome c; (ii) at each non-terminal node, easy arithmetic calculations and comparisons.

6. CONCLUSION

By weakening the independence and continuity axioms of EU theory, we have obtained a model where preferences depend only on two subjective characteristics of the lotteries: (i) the security level which they ensure; (ii) their expected utility. In this model, choice may be determined by the sole comparison of security levels; when it is not, both characteristics have an influence through a utility function V which can be set in the form

V(P) = a(Cp) Ep(u - u(cp)) + v(cp).

This expression, which resembles to Kahneman and Tversky's (1979) expression (2) for the evaluation of strictly positive prospects, suggests that decision makers evaluate separately the riskless part and the risky additional part of each lottery.

This model is more general than EU theory. By the extra flexibility of its utility function, it is able to account for Allais' paradox and similar observations; by its partially lexicographical character, it allows that compensation may not be possible in the case of extremely unfavorable outcomes, such as "ruin" or "death". Thus two of the most severe objections levelled at EU theory can no longer be levelled at this model.

This increased adequacy to actual behavior has a price - a greater complexity of the model, which is clearly disadvantageous from the prescriptive point of view. There are three functions to construct whereas EU theory requires the construction of a sole function. Yet, the amount of work entailed by the construction of these functions does not appear

C H O I C E U N D E R RISK AND T H E S E C U R I T Y F A C T O R 199

to be prohibitive. Moreover the optimization methods of decision analysis seem transposable to this model with but a few minor modifications.

The model proposed therefore constitutes a promising alternative to EU theory, should the use of this latter be judged unacceptable for theoretical or practical reasons.

N O T E S

According to Lopes (1986), some people are motivated by the desire for potential rather than the desire for security, although security motivation is the far more common pattern. The model presented could he easily adapted to suit this alternative case. 2 It is of course not excluded that both the security factor and the diminishing marginal utility of money contribute to make the decision maker risk averse. 3 This property was pointed out to me by Mark Machina. 4 It is assumed here that the decision maker is "sophisticated" (see Hammond (1976)).

R E F E R E N C E S

Allais, M.: 1952, 'The foundations of a positive theory of choice involving risk and a criticism of the postulate and axioms of the American School' (From 1952, French version). In M. Allais and Hagen (Eds): 1979, Expected Utility Hypotheses and the A Ilais Paradox, D. Reidel, Dordrecht, 27-145.

Chew, S. H.: 1981, 'A Mixture Set Axiomatization of Weighted Utility Theory', Unpublish- ed Manuscript, U. of Arizona.

Ferguson, T. S.: 1967, Mathematical Statistics, Academic Press, New York. Fishburn, P. C.: 1982, The Foundations of Expected Utility, D. Reidet, Dordrecht. Gilboa, I.: 1986, 'A combination of Expected Utility and Maxmin Decision Criteria', Tel

Aviv University, Working Paper # 12-86. Hammond, P. J.: 1976, 'Changing Tastes and Coherent Dynamic Choice' Review of

Economic Studies 43, 159-174. Hershey, J. C., Kunreuther, H. C., and Schoemaker, P. J. H.: 'Sources of Bias in

Assessment Procedures for Utility Functions', Management Sci. 28, 936-954. Kahneman, D. and Tversky, A.: 1979, 'Prospect Theory: An Analysis of Decision under

Risk', Econometrica 47, 263-291. Lopes, L. L.: 1986, 'Between Hope and Fear: The Psychology of Risk', Advances in

Experimental Social Psychology (in press). Machina, M. J.: 1982, ' "Expected Utility" Analysis without the Independence Axiom',

Econometrica 50, 277-323. McCord, M. R. and de Neufville, R.: 1986, ' "Lot tery equivalents": Reduction of the

certainty effect problem in utility assessment', Management Sci. 32, 56-60. Quiggin, J.: 1982, 'A Theory of Anticipated Utility', Journal o f Economic Behavior and

Organization 3, 323-343. Segal, U.: 1984, 'Non linear Decision Weights with the Independence Axiom', UCLA

Working Paper # 353.


von Neumann, J. and Morgenstern, O.: 1944, Theory o f Games and Economic Behavior, Princeton University Press, Princeton.

Yaari, M. E.: 1987, 'The Dual Theory of Choice Under Risk', Econometrica 55, 95-115.

University o f PARIS VI Laboratoire d'Economdtrie 4, Place Jussieu, 75005 PARIS

choice under risk and the security factor: an axiomatic model

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