Decision weights and the normal form axiom

211

Economics Letters 32 (1990) 211-215 North-Holland

DECISION

WEIGHTS

Christopher

MAXWELL

Charles River Associates,

AND THE NORMAL

FORM

AXIOM

*

Boston, M,4 02116, USA

Received 15 June 1989 Accepted 31 July 1989

In ‘decision weight’ theories of behavior under uncertainty, individuals choose between uncertain prospects to maximize a weighted average of utilities, where utility is defined over outcomes and decision weights depend upon the probability distribution of outcomes as well as, perhaps, other characteristics of the uncertain environment. The ‘Normal Form Axiom’ (NFA) requires that an individual be indifferent between two prospects which assign identical total probabilities to respective outcomes. Under NFA, decision weight theories reduce to Expected Utility theory if decision weights depend only upon the probability distribution of outcomes and not, in particular, on the nature of the uncertain outcomes.

1. Introduction In light the substantial empirical evidence that expected utility theory [see von Neumann and Morgenstern (1953) and Savage (1954)] performs questionably as a positive theory, there has recently been much interest in developing alternative theories more in accordance with behavior under uncertainty [see Schoemaker (1982) and Machina (1983) for recent surveys]. One branch in this developing literature might be called ‘decision weight’ theories [see Fellner (1961) and Edwards (1962) for early work in this area]. In these models, decision makers act so as to maximize a weighted average of utilities, where utilities are defined over outcomes, and decision weights depend upon the probability distribution of outcomes as well as, perhaps, other characteristics of the uncertain environment. In Bernard (1974) Handa (1977) and Kahneman and Tversky (1979), the decision weights associated with individual outcomes depend only upon the respective probabilities of the specific outcomes. In Karmarkar’s theory of ‘subjectively weighted utility’ (1978, 1979) decision weights depend upon the entire probability distribution, while in the theory of ‘weighted utility’ [Chew and MacCrimmon (1979), Chew (1983), and Fishburn (1983)] decision weights depend upon the nature as well as the probability distribution of outcomes. Finally, in Quiggin (1982) and Yaari (1987) decision weights depend upon the cumulative distribution function evaluated at the respective outcome levels as well as the probabilities of the specific outcomes, so that decision weights essentially depend upon the probability distribution and ordinal nature of outcomes [also, see Green and Jullien (1988) which contains the theories of Quiggin and Yaari as special sub-cases]. Many of these decision weight theories have had success in explaining certain stylized facts about decision making under uncertainty, such as the Allais paradox, the Ellsberg Paradox, and the common ratio effect [see the papers cited before as well as Quiggin (1985) and Segal (1987a, 1987b)]. * This paper was written while the author was an Assistant Professor at Boston College

016%1765/90/$3.50

0 1990, Elsevier Science Publishers B.V. (North-Holland)

212

C.

Maxwell / Decision weights and the Normal Form Axrom

In this note, I examine the impact of the ‘Normal Form Axiom’ (NFA) on decision weights. NFA requires that individual preferences depend only upon the total probability of respective distinct outcomes, so that, for instance, an individual will be indifferent between receiving x for sure and facing a lottery in which every outcome is x. Acceptance of NFA is implicit in decision theories grounded in preferences over cumulative distribution functions. I show that if decision weights depend only upon the probability distribution of the outcomes, and ignore the nature of the outcomes, then NFA will be satisfied iff the individual is an expected utility (EU) maximizer (decision weights must equal the respective probabilities of the individual outcomes). Thus, the theories developed in Fellner (1961) Edwards (1962), Handa (1977) Bernard (1974), Kahneman and Tversky (1979), and Karmarkar (1978, 1979) reduce to EU theory under NFA. I start by outlining ‘fully weighted’ utility theory, in which decision weights depend upon the entire probability distribution of outcomes. 1 then consider the ‘Normal Form Axiom’ and prove that fully weighted utility theory reduces to expected utility theory if this axiom is satisfied.

2. Fully weighted utility theory Let X denote probabilities, {GP}

the set of outcomes,

= ((x1,

xz,...,x,);

and define

(Pl,

a ‘prospect’

P2>...,PJ},

to consist

of vectors

of outcomes

and

(1)

where x,cXVi, and PEA”-’ (the (n - 1)th dimensional simplex), so that p, is the probability that the outcome will be xi. The ‘fully weighted’ utility theory framework requires that {x; p } is weakly preferred to {a;j?} iff V{x;p} 2 V{$@), ’ where

V(x;p)

= i: $(P>WX,>, 1=1

(2)

and where r,“(.): A”-’ -+ lR+ and U(.): X+ R. ’ The v,“( p)‘s are the ‘decision weights’ assigned to outcome x, and may, in general, depend upon the entire probability distribution p. Note that in some of the theories above, decision weights depend only upon the specific probabilities and not, in general, upon the entire distribution.

3. The Normal Form Axiom Define associated

the ‘total’ probability of an outcome in a prospect to be the sum of the probabilities with that outcome in the given prospect. Consider the following axiom:

Normal Form Axiom (NFA). Consider two prospects, and suppose it to be the case that whenever the first prospect assigns total probability q > 0 to a specific outcome, the second prospect does so as well. Then and individual will be indifferent between these two prospects, In other words, given NFA, an individual will be indifferent identical positive total probabilities to respective outcomes. individual assigns no value to zero probability outcomes. ’ A is ‘weakly preferred’ to B if A is ‘at least as good as’ B. ’ Iw+ is the set of non-negative real numbers.

between two prospects which assign In particular, NFA implies that an

C. Maxwell

If NFA (a)

{(xi,

is satisfied, x2);

/ Decision weights and the Normal Form Axiom

then we have the following

(p,l-P)}

- {(xi,

xl, x2);

indifference

(p-4,

213

relationships:

4,1-p)},

Vx,&x,,andVpE[OJ]

(b) {h

[&PI, x1): (P, 1 -P>} - {(x1); (I)}, vx,, and VPE (0, 11,

cc>

{(x17

x2);

(l,O)}

(4

{(xl,

x2);

(P,

and q E

-

{(x1);

1 -P>}

-

(I)},

((x2,

vxl

xl);

&x2,

(1

-P,

and

P)},

vx,

&x2,

and

VPE

[OJI.

In NFA(a), the probability of xi (x2) is p (1 -p) in both prospects. NFA(b) implies that the decision maker is indifferent between receiving x1 for sure, and a prospect in which all outcomes are x,. The fact that no value is received from zero probability outcomes is captured in NFA(c). NFA(d) indicates that preferences do not depend upon the particular ordering of outcomes (and respective probabilities) in a prospect. Notice that NFA rules out state-dependent utilities, and is implicit in theories which assume that uncertain situations are completely characterized by the associated cumulative function. Proposition 1. If a fully weighted utilit_v maximizer values prospects according to (1) and if this individual also satisfies NFA, then this person must be an expected utility maximizer: r,“(p) =p,, Vi, n, andpc A’-‘. In Handa (1977) utility is linear and decision weights depend only upon the specific probability of respective outcomes. Fishburn (1978) has shown that Handa’s theory reduces to expected utility theory under a mild ‘dominance’ axiom. Proposition 1 can thus be viewed as a generalization of Fishburn’s results for cases in which decision weights depend upon the entire probability distribution of outcomes. Proof. The proof is in three parts. I first show that NFA implies that without loss of generality, ~~((1)) = 1. Following the argument in Quiggin (1982), I then show that given NFA, r,,“(p) = f=,p,) -f(C,l:p,)Vi, n, and p E A’“-‘, where f(p) = T:( p, (1 - p)). Finally, since 1 find that given NFA, f( .) must be a linear operator, r,“(p) =p, Vi, n, and p E A’“‘. Notice that I assume a certain minimal richness of the domain of prospects: There are at least three distinct outcomes over which the decision maker has a strict preference ordering, and preferences are defined over all lotteries in which at least one of these outcomes occurs with positive probability. (i) NFA(b) above implies that Vx and Vp E An-‘, {(x, Cy= ,7~,~(p)U(x) = ~~((l))U(x). Since U(x) # 0 for some x, generality, normalize the decision weights so that ~~((1)) = 1. (ii) 3 Define f(p) = TT~(p, (1 -- p)). Given the normalization NFA(a) implies that this person must be indifferent between L, = {(Xi,

x21;(pl

+p2,

p3)),

and

Thus, letting p = (p,, p2, pj), it must therefore p2))U(x,) = [r:(p) + 7~2(p)]U(x,) + T:( p)U(x,). V:(P)

+$(P>

’ The proof in the section

=f(:pl

follows Quiggin

+PZ),

and

(1982) closely.

L2=

{lx,,

x,. . ., x); (p)} - {(x); (l)}, C:=,T,~,“(p) = ~~((1)). Without

so that loss of

of weights, 7~2(p, (1 - p)) = 1 -f(p). the following two prospects: xl,

x2);

(P,,

pZ, P,>>.

be the case that f(p, +p2)U(xl) + (1 -S(p, Since the x,‘s are arbitrary, this means that T?(P)

= 1 -f(~,

+p2).

+

(3)

C. MaxweN / Decision weights and the Normal Form Axiom

214

Further, since NFA(a) -% = ((-5

also implies that this person must be indifferent between ~~1; (pt,

p2+p3)),

and

it must be the case that f(pt)U(xi) +f(p2 Since the xi’s are arbitrary, this implies that n13(PI =f(

PI>>

and

T?(P)

L=

{(xi,

+pl>U(x2>

+ r:(p)

x2, ~~1; (pi,

= r:(p)U(x,)

p2, P,>>,

= [v&p)

~2 t-p,).

=f(

+

77~(p)]U(x,>.

(4)

Eqs. (3) and (4) yield (5)

(f( p, fp,

+p3)

= 1 given our normalization).

T”(P) =f ( j=li

P ,)

(iii) Finally, let a(.): {1,2 ,..., that this person must be

-~(XP,)

Applying induction, we find that

vi,

n,

and

PE

(6)

A”-‘.

n} --) {1,2 ,._., n } be a permutation operator. Then NFA(d) implies indifferent between {(x,, x2, . . . , x,); (pl, p2, . . , p,)} and

{(X,(I), X,(2), . . .T Ql,>; (Pm(l)> PO(Z)>. . .? Pod}. Since this must obtain for all prospects (and since outcomes can be independently varied), More specifically, this requires that n&( pl, p2, p3)) = n,“(P) = 7i,;;)(j), where pi =jo(;). n;(( p2, pl, p3)), which, given (4), yields f( p, + pz) - f( pI) =f( p2). f( -) must therefore be a linear operator. Given (6), then

n,“(p)=f

ii_,iP

ii

-/(KP,)

= &CP,)

-

Xf(PJ

=_f(P,)-

Notice that the 7T,n(p)‘s do not depend upon the entire probability distribution and are, instead, solely a function of the respective probabilities. Since f(p) = a + pp, Cf( p,) = na + /3 = 1 V n * (Y= 0 and p = 1. Thus, f(p) = p, and we have expected utility theory. 0

4. Conclusion If the Normal Form Axiom is accepted, then the results in this paper suggest that decision weight theories must allow decision weights to respond in some fashion to the nature of outcomes. This specifically rules out decision weight functions which depend only upon the probability distribution of outcomes. Of the decision weight theories discussed at the outset, Chew’s (1983) ‘weighted utility’ theory is compatible with NFA, as are the theories of Quiggin (1982, 1985) Yaari (1987), and Green and Jullien (1988) in which decision weights respond to the ordinal nature of outcomes.

References Bernard, G., 1974, On utility functions, Theory and Decision 5, 205-242. Chew, S., 1983, A generalization of the quasi-linear mean with applications to the measurement of income inequality and decision theory resolving the Allais paradox, Econometrica 51, 1065-1092. Chew, S. and K. MacCrimmon, 1979, Alpha-nu choice theory: A generalization of expected utility theory, Unpublished manuscript (University of British Columbia, Vancouver, BC).

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215

Edwards, W., 1962, Subjective probabilities inferred from decisions, Psychological Review 69, 1099135. Fellner, W., 1961, Distortion of subjective probabilities as a reaction to uncertainty, Quarterly Journal of Economics 75, 670-699. Fishburn, P.C.. 1978, On Handa’s ‘new theory of cardinal utility’ and the maximization of expected return, Journal of Political Economy 86, 321-324. Fishburn. P.C., 1983, Transitive measurable utility, Journal of Economic Theory 31, 293-317. Green. J. and B. Jullien, 1988, Ordinal independence in non-linear utility theory, Discussion Paper, no. 7380, May (Harvard University, Cambridge, MA). Wanda, J., 1977, Risk, probabilities, and a new theory of cardinal utility, Journal of Political Economy 85, 97-122. Kahneman, D. and A. Tversky, 1979, Prospect theory: An analysis of decision under risk, Econometrica 47, 263-291. Karmarkar, U.S., 1978, Subjectively weighted utility: A descriptive extension of the expected utility model, Organisational Behavior and Human Performance 21, 61-72. Karmarkar, U.S.. 1979. Subjectively weighted utility and the Allais paradox, Organisational Behavior and Human Performance 24. 67-72. Machina, M., 1983, Generalized expected utility analysis and the nature of observed violations of the independence axiom, in: B. Stigum and F. Wenstop, eds., Foundations of utility and risk theory (Reidel, Dordrecht/Boston, MA). Quiggin, J., 1982, A theory of anticipated utility, Journal of Economic Behavior and Organization 3, 323-343. Quiggin, J., 1985, Anticipated utility, subjectively weighted utility and the Allais paradox, Organizational Behavior and Human Decision Processes 32. Schoemaker, P., 1982, The expected utility model: Its variants, purposes, evidence, and limitations, Journal of Economic Literature 20, 529, 563. Segal, U., 1987a. The Ellsberg paradox and risk aversion: An anticipated utility approach, International Economic Review 28, 175-202. Segal, U., 1987b, Some remarks on Quiggin’s anticipated utility, Journal of Economic Behavior and Organization 8, 145-154. von Neumann, J. and 0. Morgenstern, 1953, Theory of games and economic behavior, 3rd ed. (Princeton University Press. Princeton, NJ). Yaari. M.E., 1987, The dual theory of choice under risk, Econometrica 55, 95-115.