Mathematical Social Sciences 1 (1981) 155-168 Q North-Holland Publishing Company

GENERAL CONDITIONS FOR NON-NORMALITYOFRlSKAREAS

Communicated bv l%cd W. Rowh Rewivcd 12 lG3mury 1980

Let R, bc the option with brings success with proballility p and failure with probabilit> I - p, and Ict R’ be the tisklcss option. The risk area is the set of probdbilitics y for which R, is better than R’. A risk area A is said to be normal, if p E A and p’ > p imply p’ E A. The paper SLOWS that under very general conditions the nownormA risk areas will appear (for suitable V~UCS of rcwrds and losses) in all classes of situations, except those covered by the SW nwdcl.

KCJWW& Normal risk area. Subjective expected utility. Decision maker.

1. introduction

The aim of this paper is to present a certain new decision model, designed primarily tcj

COWS those cases which are not covered by the classical SEU (Subjective Expected I ItiLt>*) model [see Coombs (1970); Davidson (I 957); Edwards (I 961): Krant z ( 1970)], and at

the same time, to incorporate in the model the psychological features of the motivation model of Atkinson (1964). ( 1970): Coombs ( 1970); Heckhausen ( 1973).

The model will concern binary choices, in which the decision maker is to choose be- tween a risky option R and a riskless option R’. The latter involves no uncertainty. and brings the reward 6. It serves therefore mainly as a reference point: option R ~‘111 be chosen, ifits expectation E(R), to be defined later, exceeds the utility I#) of the riskless option R’.

The option R may lead to one of the two outcomes. labeled succ’ess and failure. with (subjective) probabilities p and 1 -- 11. The success brings reward L and f’llilure leads to a loss tl (or: brings ‘reward’ - d).

Thus, graphically, the considered decision situation may be represented in time o t’ ~1 tree. as in Fig. 1.

The value-however defined-of the risky option R, must be a function of at kast the following three parameters: (1) the subjective probability ~1 of success. i 3 the amount r of reward for success, and (3) the amount d of loss in case of faiiure. The main object uf interest will be the dependence on the first of these parameters. Thus. rmernberillg that

1%

I’ and (1 are also intervening in the nIod4, we sh3ll denote by R, the risk\: option ~h~ll3C

terized by the probubilit y I:, of success. Let 9 stand for the relation ‘is better tlIiIl1.‘

Definition. The set of all 11 E [ 0, I 1 such t h3t R, l > R’ will be called risk ~VJ.

Definition. A risk 3re3 A C (0, I ] is c;dlcd mwwl, if

01’ course, p1 in the above definition must belong to the interval (0, 11; in geneml. it will 9e tacitly assumed in the sequel that all numbers appearing as probabilities are so restricted.

Thus, a risk area will be not nornx& or non-normal, if t’or some p1,~2 with p1 <pz. we havey, EA andP2 $A.

Among non-normal risk areas, we shall distinguish two types: cwtml3nd bowdur,v. A non-normal risk arc3 is said to be central, if for some p1 <pa < p3 the rmiddle member belongs to it, while the two cxt reme members do not. A risk are3 is of boundary type. if together with e3ch of its points, it contains all points less than it (hence boundary risk areas consist ut’ an interval starting at p = 0, and are therefore, in 3 sense, dual to normal risk areas). Needless IO say, central and boundury risk 3~3s do not exhaust 311 possibilities for non-normal risk areas, since it is not required that a risk urea be connected.

Let generally f’(p) stand for the value (expected value) of opt ion I$. It is clear that all risk areas will be normal if, and only i&f’(p) is monotone increasing.

At first sight, normal risk areas seem most natural: the more likely is the success, the more ‘profitable’ shculd appear the risky option R, (other conditions being equal).

On. the other hand, non-normal risk areas might appear as somewhat anomalous: with such a risk area, the decision maker would choose a risky option R, for some value 3, but

157

if the chmxs of success were increased. he would choose the risldess option R'. One might he tempted to !sbel such a behwiour as ‘irrational.’

One d the nlsin results of this paper. however, is that (except tb;_ tlw extreme case u! the classical SEU model), non-normal risk areas are rather a rule thm an exception. htore precisely, except for the SEU model. non-nornlal risk areas will appear in my sit ustion. for suitrrblc values of rewards, losses. and conlp;lrison level 6.

It is cwntial to realize at this point that whether or not non-norm1 risk areas rw~~- appear depends only on the behaviour of the value function f’(p). while Acther they rr-ill actually appear (if they may) - depends on the wnlparison lewl. This is illust rrltai in

Fig $ at which j’is not monotone. Thus, the non-normal risk areas my appear. mi the!, 8 0.

actually do appear fx comparison level k. and dc) not appear for k’. To summarize: the main thesis of this paper is that there esists ;t class of situations.

und a psychologically defcnsibk definition of the value functionf~p) of risky option R,. such that for some parameters r and tl, the function]‘(p) is not nwnotonicall~~ increasing. Thus, for such values of the parameters, the non-normal risk areas nwt appear for appro-

priate comparison levels.

2. General postulates

We shall wcept the following:

158 M. Nowako wsh / Geneml corrditions fiw tiownorrrrality of risk areas

j’(p) = value of R, = pv(s) + (1 -- p) v(S’) ,

where v(s) and v(S’) are values of sucves~ mtd failure.

Let us see first what is the situation in the classical SEU model. This model is charac- terized by the crucial assumption that the values v(s) and v(S’) of success and failure do not depend on probabilities of their occurrence. Thus, v(s) depends only on the reward r. and is customarily referred to as utility u(r) of the reward r. Similarly, v(S’) depends only on the amount d of loss, and is equal to the utility u(4) of loss. Consequently. relation (1) reduces to

f(p) =pu(r) +(1 - p)u(-d) =p[u(r) -- u(--d)J + u(4). (3

Implicit in the terms ‘success’ and ‘failure‘ is the assumption that 1((r) > I+-&, so that j’(p) is an increasing (linearly) function of p. We have proved thercforc

Theorem 1. lrt the classical SEU model, all risk areas ate ncmttal.

The 1> !sic postulate of the SEU model, of independence of utilities of success and failure of their probabilities, may be reasonably expected to hold in cases when the deci- sion maker has no control over the probabilities of occurrence of success and failure.

However, when the occurrence of success depends - even partially - on the decision maker, namely on such of his features as skill, knowledge, persistence, etc., then the assumption of independence of values of v(S) and v(S’) on p is no longer adequate. In such cases, one may expect that the leqs likely is the success, the higher satisfaction from its occurrence.

Thus far, the only model which took such a dependence into account was the motiva- tion model of Atkinson (1964,197O). However this model, in turn, neglected all the ‘economical’ variables, and concentrated mainly on the choice of task difficulty.

Incidentally, one could argue that even in the case of total lack of control of the deci- sion maker over the probabilities, the values v(S) and v(S’) may depend on p. Imagine a person who wins a prize in a lottery, say a color TV set. He comes home to tell his wife about the event, and says: “Can you believe that 1 won this prize, even though there was only one chance in 10 000 (say)?“. Perhaps he would be less elated if the chances of win- ning the same colour TV set were (say) one in a hundred - since there would be no factor due to something ‘unusual.’

3. The general model

Let us now consider a general choice model outlined in the preceding sections. Accord- ing to the main postulate, the value of the risky option R, is the weighted average of val- ues of the t.wo outcomes.

We shall assume generally, that the value v(S) of success depends on three factors:

( I ) amount r of reward; (2) probability p of success, and (_I) strength .I! ot‘ the motive to

achieve success. On the other hand. the value u(f) of failure will also depend on three factors:

( I ) amount d of loss; (2) probability p of success (or alternatively: probability 1 - p ot failure), and (3) strength F of fear of failure (motive to avoid failure).

In each case, the first two fktors present no difficulty. Some explanations. however.

is required ffbr the motives M and F. These notions are due to Atkinson ( 19&l); their intu- itive content is quite straightforward, and an infornr I! definition is rather clear. Thus. M is tire general tendency of the decision maker to achiew success. pkliding his behaviour towards situations in which achievement is possible. 0 the other hand, fear of f‘ailure 1. paralyses his actions, causing him to withdraw from rid Y siturltions which may lead to f3ilure.

The main problem consists of measurement ot’ these I lotivcs. Since the exact values

are, thus far. beyond reach. and one has only tests which provide ordinal data. the model below will be constructed so that the lack of knowledge of esact values ofM and F will not prevent ane from making qualitative predictions.

It will be assumed that Mand Fare some positive values, which enter xnultiplicrttivel~ to the values u(S) and ~($1. Thus.

v(s) = Ml,(P. r) * v(S’) = F&f(p. ci) .

The function &(p. r) will be called the incerdw of‘success. while laf(p. ‘1) ~i.ll be dd

the irwerrtivc to awid jbilure (see At kinson ( 1964)). We shall now formulate the hypotheses about these functions.

Id P* r) = C(P) + H(r) ,

C(P) -+ 0 asp-4,

G’(p)+c<O asp3 1.

H(r) -+ 0 iis r -90.

Condition (a) of Hypothesis 1 is of technic;ll character. md scarcely requires any CWI-

merits: it asserts thst the rate, rrt which the incentive ot‘ suc~css chmges with the chmgc of probability of success p remains bounded. Observe that this condition does not assert

anything about t!le sign of the derivative with respect to p.

Condition (b) of Hypothesis 1 states generally that the variablesp and r in the incen-

tive I,(p, r) of success separate additively in the neighbourhood of the point p = i . r = 0.

160 M. Nowvlkowsku ,/ General conditions for non-norm&y of risk areas

It says therefore that the psychological and economical effects are additive. at least in the donr;rin of sd.l rewards and high!-* probable sucesses (i.e. in the domain where these effects are weak).

In still other works, this assur+ion states that p and I do not interact in the incentive value of success (in the neighbou,:lood of p = 1, r = 0).

The justification of the three specific assumptions (S), (6) and (7) is as follows. The incentive of success is (or may be) connected with overcoming some difficulties, use of skill, knowledge, inventiveness, etc. In any such case, the less likely is the success, the more satisfaction brings its occurrence, which means that I,(p. r) should decrease with the increase of p, hence its derivative with respect top should be negative.

In fact, Hypothesis 1 requires that the incentive of success should decrease with p only in the neighbourhood of p = 1 (condition (6)); no monotonicity requirement is imposed for other values of p.

Interpreting the functions G and H in condition (4) as the effects on incentive of suc- cess due respectively to the probability of success (hence from the occurrence of success ‘as such’, regardless of the economical reward r), and to the economical reward r, one may say the following. Condition (5) states that the satisfaction from ‘success as such,’ sepa- rated from the economical reward, becomes negligible when the success becomes certain. However, the rate of change of this effect asp tends to 1 is not negligible (condition (6)): as p bet bmes larger, the psychological satisfaction from success becomes smaller at a rate which approaches some negative value.

Finally, condition (7) asserts that the effect on the incentive of success from the reward r becomes negligible when the reward approaches zero.

The following assumption about I,(p, r) will be needed in one of the theoren~s only; this explains why it is listed separately.

Hypothesis 2. If r > 0, therz

lim I,(p, r) = c, > 0 . (8)

Intuitively, this hypothesis asserts that the incentive of a success which is very improb- able, and which brings a positive reward, is positive. A moment of reflection is sufficient to realize that condition (8) is very natural and plausible.

Let us now turn to the hypothesis about the incentive value I&p, d) of avoiding failure. It appears that for t1.c existence of non-normal risk areas, the assumptions about the func- tion I;if arc still weaker than those about the function IS. Again, we separate the condition which is needed in one of the theorems only.

Hypothesis 3. The functim &f( p, d) is differmtiizblr with respect to p ,/br evecv d, ad the derivative al,#p is bouncded itI the closed irtterval [0, 11.

Hygothesis 4. For all d > 0 we have

lim I,&, d) = qd < 0 . P-1

(9)

161

Both of these hypotheses hardly need any intuitive just ificatron. Hypothesis 3 is again

of a technical character. asserting simply a certain degree of smoothness of function I,@,& Hypothesis 4 states that for p sufticicntly close to 1. hence fur failures which art’

highly improbable, the values I,f(p, c/) are negative (at least for failures which result in a loss d > 0). To put it differently: if cd > 0, the failure always ‘hurts,’ and it hurts most when it is improbable. This, incidentally. suggests that a natural requirement on &r would be that it is a decreasing function ofp. We do not state it 3s an assumption, however. since it will not be needed in establishing the proof of the theorems.

We shall now prove two theorems about the ex%tencc of non-normal risk areas.

(hf > &) * ( 3@( Vr) [(r < ro) * (3 b): risk area is r~uwwrmalj .

What this theorem asserts, in essence. is that if the motive M to achieve success is suffi- ciently high, and the reward r for success is sufficiently small, then one can find such ;1 reward for the riskless option R’ that the risk area will be non-normal: the decision maker will prefer the risky option R, for some p, but when the chances p of success are increased sufficiently close to 1, he will prefer the riskless option R’.

Roof. It suffices to show that the value function f*(p) of the risky option R, is not mono- tonically increasing. For that. in turn. it is sufficient to prove that there esists 9 point at which the derivativeI” is negative. We shall prove that such 3 point exists in the rkgh- jourhood of p = 1.

Using (1) and (3) we obtain

(10)

Let us now choose p close to 1 and r close to 0, so that the relation (4) of Hypothesis 1 holds. Substitution of (4) into (10) yields

f(p) = pM(G(p) + H(r)] + (1 - p) F&,f(p. 4 .

Next, let us fix F and [I, and let

MO = F&t/c .

(11,

(13

where t’ is the constant appearing in (h). and 11~ is th: limit in condition (,‘I) (it is nut required now that qd be negative). Since (* < 0 by assumption (6). we have for all M > .I&_,

MC- Fq&O. (13

162 M. Nowko wska / General conditions fbr non-normality of risk areas

Differentiation of (11) gives

f’(p) = MG(p) + pM(;‘(p) + MH(r) -- Fl,dp, d) t (I --- 11) F ‘2 . (14)

LRt us now fix M>MO and pass in(14) to the limit withp --+ 1 andr l 0). We have then MG( p) + 0 by (S), MH(r) + 0 by (7), and (1 - p) F&/ap + 0 by the assumption of boundedness of the derivative of I,f (Hypothesis 3). Next, pMG’(p) + MC by (6). and finally, Fl,f(p, d) -+ Fqd by (9). Consequently, the limit of the derivative (14) asp + 1

and r -+ 0 equals MC - Fqd, a quantity which is negative in view of (13). This completes the PrOOf.

We shall now prove

Theorem 3. IL it? additiorr to the assurnptims of* lheotem 2, the iricerttiws I, arrd l,f satis- f‘j? respectively Hypotheses 2 and 4, thert me cat1 jhd values F arid d > 0 such that 1;~ suitable b, t/w /isk area is t1m-mwn1~1 artd cerltml.

Proof. 3y the preceding theorem, the value functionf(p) is not monotonically increasing: it decreases in the neighbourhood of the point p = 1 for suitable values of the parameters. To prove the assertion of &he theorem, it suffices to show thatf@) is not monotonically decrea&g in the whole interval [0, 11. We shall show that for suitable choice of F and (i, the derivative f’(p) is positive in the neighbourhood of p = 0.

LRt us first fix F0 and do > 0, choose MO defined by (12) for (F,,&), and then choose M > MO and r such that f’(p) is negative in the neighbourhood of p = 1, according to Theorem 2.

Let us now differentiate (10); we get

f'(p) = M&(p, r) + pM a+ - Fl,dp, d) + (1 - p) F ‘$ .

Let c, > 0 be the value of the limit in (9), and let 0 < e < Mc,./3. From relation (9) we have then I&p, r) > c, - r/M for all p sufficiently small, say p < po. Next, since the deriv- ative a&/ap is bounded (Hypothesis l), one can find p1 Gpo such that for all p <pl we have JpMal,/ap I < e. Consequently, for p < p1 we may write

= Mcr -2c+FQ, (16)

where, for simplicity, Q = (1 - p)(U&p) - I&p, d). In view of the condition on e, we

get

f'(p) >Mc,/3 + FQ , (17)

and the last value is positive for sufficiently small F, in view of boundedness of the deriv-

at ive i&#p, guaranteed by Hypot heeis 3. To complete the proof it sutt‘iccs to observe that the decrease of Fdoes not case titq

violation of conditions implying th3t J”(p) < 0 for some p 40s~ to I : indeed. under the assumption cl > 0 of the theorem, and under Hypothesis 4, if condit ion ( 13) holds for some FO. then it also holds for all fX &,. This completes the proof.

The theorems of this section show that non-normaal risk 3re3s 3re r3Wr 3 rule than an exception: under some very general and quite plausible asmnptions described by Hypoth- eses I “- 4, one should observe non-norm;ll risk areas for suitable comparison levels, at least for persons with high achievement motive M, and in ~3s~ of sr113ll rcwxds r.

The SW model is an exception here; yet, man) people are trained tu think in terms of SEU model, ayd accept it as a standard of rationalit_.. In this saw, the thsowms of th!s section show that apparently irrational behaviour 1113. * in fxt bc easily explained by the concept of non-normal risk areas.

4. some special ows

In this section we shalt discuss briefly some plausible csses of v3lue functbtls anti

resulting risk areas. Firstly. let us observe that the original model of Atkinson ( 1970) for ad~ievment

motivation was obtained by putting&, = 1 - p. 13f = -p (so that the rc\vards r and losses tl were not taken into account). We have then

which is 3 pnr3bol3, with bnnches going upwards or downwards depending on the sign ot‘

the difference M - F, and with the extremum at p = l/2. In each of the r’ascs ill > F 2nd

hf CF the suitable choice of comparison level b yields 3 non-nortnd risk ma. For certain interesting implications of this model to the problem cjf dc~ision rnwwnts

in scientific careers, see Nowakowska ( 1975). The model of Atkinson concerned the extreme case. when the incentives ci+nd O~I$

on probabilities (i.e. on task dift’lculty). In 3 somewhat more general case. one wuld pas- tulate

and

v(S’) = FI.,f = -I$ - cl .

where far simplicity, we assumed that the JCW~J~S 3mI Iosst’s are expressed in ~mlit>~ units.

so that we can write simply r and d instead 01’0 (r) 311d U( ~0.

We have then

Let us find conditions under which all risk areas are normal. In case when M > F, ix. wkn the motive to achieve success dominates over the fear of’ failure. the branches of (he parabola (2 f ) are directed downwards, and j’(p) will be monc)tonically increasing in the interval IO, 1 ] if, and only if, the maximum occurs to the right from the point 11 = I. This means that we must have

r+d+M- F

2(Al- k3 -w, (22)

which reduces to

r+ii~AI-- F. (23)

If M < F, i.e., when fear of failure is st ronger than the motive to schievc success, the branches of the parabola (2 1) are directed upwards, and j’(p) is increasing in the whole interval (0. 11 if, and only if. the minimum occurs to the let1 from the point JI = 0. This means that we must have

(24)

which reduces to r+d>F--hf.

WC 11 .ty therefore formulate

Theorem 4. If the values of srrcwss aird $3ilwc are girl by (I 9) arid Q9). tlwr d risk areas arc itmmal ui and wt& iJ’:

rtd>iM-- FI. w

This result may be interpreted as follows. If the rewards r and/or lossesd are large, then all risk areas are normal.

Alternatively, non-normal risk areas may appear (for suitable comparison levels) only if both the rewards r and losses d are sufficiently small. The valde of this result lies, among others, in its qualitative .character, largely independcalt of the values M and F, which are, as already mentioned, unavailable empirically.

Conditions (19) and (20) postulated that the incentive functions & and &f depend on the motives M and Fz we have here Is = 1 - p + P/M and I,,r = -p - d/F. It is of considers able interest to abandon this assumption and postulate

I,(p, r) = 1 -- p + r , I,f(p,d) = -p -- cl , (27)

(see [ 12 j and [ 131). Substituting into j’(p) yields again a parabola, with -(M - #I as a coefficient at p2. However, the remaining coefficient now also depend un M and F.

The detailed discussion of risk areas, and their dependence on particubr parameters. may be found in the quoted papers Aowakowska (1979a). (1979b). as well as - for sonic special cases of interest -- in Nowakowska (1980). Paper [Now crkowska (19X)] presents implications of the theory for the case of decisions concerning innovations.

166 M. Nowzko wska / Gowtal mnclirions for non-notmlity of risk atcas

in which he is to make the choice of one of the two options (the generalization to the casC of nlore options is obvious). The decision situation is described by a number of paramc- ters; in the present case, the Farameters are p, r, tl and b. In addition, the decision maker hinself is also characterited by some parameters (in the present case, the motives M and b].

Combining all the parameters together, one obtains a representation of a particular decision problem (together with the decision maker) as a point in a multidimensional Euclidean space.

In the present case, wz have therefore six parameters. Consider generally. a class of decision situations with I: opt ions, RI, . . . . Rk, each member of the class characterized by a multidimensional paramtbter S= (~1, l . . . s,,). the coordinates being losses, rewards, probabil- ities, motives, etc. Thus, FE P, and let S C fi* denote the class of all admissible situations, ;IS determined by various constraints on coordinates of S-(e.g. that all probabilities be between 0 and 1, etc.).

A solution of the class S of decision situations is a family of k sets. SI. Sa, . . . . Sk such tha’ SI U S2 U l U Sk = S, with the interpretation being such mat if SE Si, then the best choi e in the situation described Sy Sis the option R,L

Observe that it is not required that the sets Si be disjoint: if FE Si n Si, then the choice of option Ri is as good as the choice of option Sj.

Let us note that the above is the weakest concept of a solution, since it is required merely that we know the best options, but not their numerical valuations, nor even their ranking: q 3 could, of course, define solutions in stronger sense, e.g. by requiring that to each Fti,r;re is assigned a ranking of the set of options, or - still stronger - a vector of numerical valuations.

Let us observe that the conc$lpt of decision space used here differs from the notion of motivational space introduced in Nowakowska (1973). There, such a space was charac- terized separately for each individual, and was spanned by a number of dimensions (psy- chological continua, or scales, as represented linguistically by classes of special functors, e.g. cpistemic, ;uch as ‘1 know that ,’ ‘I am certain that,’ deorttic, such as ‘I ought to,’ nlofi- ~~ational, such :G ‘I want,’ ‘I prefer,’ etc.). These functors were obtained through abstrac- tiun trorn the content of sentences which one uses to explain, justify, or evaluate the deci- sious which one is planning to make, or which have been made. The linguistic representa- tion of such scales, as e.g. of subjective probability, its richness and discriminativeness, were (in decisional context) analysed for the Grst time in Nowakowska (1973).

In such a space, each option in a decision situation is represented as a point, and the problem lies in assigning a ranking of options, meeting various ‘natural’ constraints. In Nowakowska (1973) it was shown that the formal structure of individual decision is the same as that of the group decision, and in consequence, the Arrow Jmpossibility Theorem applies here [Arrow (1963)]. It turns out that such an interpretation leads tc certain psy- chological consequences concerning the process of decision making, sinjply due to the observation that people must be partitioned into categories, depending on which of the Arrow postulates (hck of dictator, sovereignity, independence of irrelevant alternatives, etc.) is broken in a given situation. Each of these categories may be, in turn, interpreted in psychologic;ll terms.

This conclusion 1‘rom Arrow Tht~rc~n is k)t‘ particular ilq>c\rt:l1\~c 1‘ro111 tilt‘ <~ylit i\,e

point 0T view. since it provides :I thtwretkal just it‘iitition ,lt‘ the t;fc’t that iri each Jccisiolj sit nation in which t hc fitiid cvrrhrrt ion is bxml on prtiill t’v;liu3t ices. it is t~cccssaq. to

resign from SWW conditions, which one wcwlc~ usu;rlly like to have as ‘natural.’ In othcl words, if one has a transitk relation of prcfcrenzrr. it is always ;!t the cost 01’ i>rcttking 3

mtain ‘niltural’ rule of aggregating przlircnces.

Mor~vet, the theoreti4 and cmpirictrl significance of the &WC ident it‘ic;rtion ot‘ ttlc

individual and group dc&ion mhinp. and application 01‘ r\rrow Thwwm lies. firstI>*. in

mkkrinp the role of mlinal psy~hol.~~~~sl cant i!jua. Thcsc art’ t c.ditionrlIIy neglected in

kision modck. wbitc in realily. Ihcy intervene in 1’~ proCt24. SKXMI~~ . these considera-

tions lead in H nmral way to the dcfini!irm d intcln. I ~ontll~:ts. rtnd 3 t;ts0n0llly d such contlicts [SW No~akowska ( t 9731. t t is worlhH bile I 1 nienti0n that ewi in the sirnpicst case of CciIltliCtS of four c’ompctinp niolivcs which c’ann ht he jointly satistlcd. the tiufiifwr

of different types 01‘ contlicr s is ;1s high 3s 1 I 3.

For empirical invesripalit IN. the stpprorlch brlsct! on a~ applim im 01‘ Arrow t heoren~

s\rgpsts (a) mdysis of C:)SWS 01‘ situativns. in which the criteria cannot bc ;ig,grq;ited intc)

;a transitive relation. and (b) in cases of ob:aininp such a relrrtian, observing which of the postulates arc broken. The ultimate pod of such an analysis would be to determine glasses of decision situations in which a given postulate would be most likely to be broken. The

knowledge of such classes would allow restriction of the class of de&m rules applimbit‘ in a given situation.

Retl.rning to the problem of the present paper. the desision space is spanned by six dimensi Ins. of which at least four correspond to scales of the interval type. or higher ( the

status of scales for reprcsentq the motives Al and F need not be resolved here). The set S

of admissible situations is defined by the constraints

o<p< 1. icl>O, F>O. r> -tl (b may be arbitrary; the last condition above is implkit in the use &terms ‘stl<<t’ss ;IIIC~ ‘failure’).

A solution to this decision problem is a set S* C S oi those vestorb s t’or \vhi<h the risk!

option R is better (or as good as) the riskless option R’. This paper concerned the study of the properties of the interswtiwl ofS* with the' sub-

space of S obtained by fixing all values of coordinates in ‘S-except t‘ur ~3: this subspxc is therefore an interval of length 1, and a non-nornlal risk area cwurs simpl! ii :!;is maul

intersects with S*. but its right-hand side end is not in S*. The most important feature of the results of this paper is their gtmrality. TIE>, ~IIM

that non-normal risk areas are bound 10 wc’ur under practkally all wmiitions u hkh Ltrc not covered by the SEU model because of the depcndenie it’ wlues d’ su~~ss ~1~1 t1lilui-< on their probabilities.

These findings have some deep psychological c’onsequen~es. The>, may e.\plairl jcjlw 01’

the observed deviations of empirical data front the predic’tiom ot‘ thy SEL’ rttdcl. S:we

the Mter is generally rhought of as a standard of rationalit> . the tltwrms prowl irl this

paper may explain certain behaviours which OIW would be irxlined to tt’rI;i ‘imti~wl.’ In

this sense. the present model offers some rationalization 01‘ irrationality.

168 M. Nowukowsh-a / Gerieml conditions for nownorrr~ality of risk areas

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