Journal of Mathematical Psychology 47 (2003) 244–258
Probability weights in rank-dependent utility with binary
even-chance independence
David E. Bella, and Peter C. Fishburnb
a
Harvard Business School, Boston, MA 02163, USA
b
AT&T Shannon Laboratory, USA
Received 28 August 2001; revised 8 July 2002
Abstract
We examine the effects of a weak version of expected utility’s independence axiom on the probability weighting function in rankdependent utility. Our weak independence axiom says that a 50–50 lottery between a two-outcome gamble and its certainty
equivalent is indifferent to the certainty equivalent. A variety of nonlinear probability weighting functions satisfy this axiom, but
most weighting functions proposed by others do not. Nevertheless, the axiom accommodates weighting functions that are quite
similar to the inverse S-shaped concave–convex functions of others that overvalue small probabilities and undervalue large
probabilities.
r 2003 Elsevier Science (USA). All rights reserved.
1. Introduction
It is widely acknowledged that expected utility (EU)
theory (von Neumann & Morgenstern, 1944; Marschak,
1950; Herstein & Milnor, 1953; Luce & Raiffa, 1957;
Raiffa, 1968; Fishburn, 1982) is descriptively inadequate
as a theory of individual choice for decision under risk
despite its appeal as a normative guide for rational
decision making. A prominent feature of some nonexpected utility theories that accommodate prevalent
behavioral violations of EU, including rank-dependent
utility (RDU), cumulative prospect theory, and close
relatives (Quiggin, 1982, 1993; Luce, 1991; Luce &
Fishburn, 1991, 1995; Tversky & Kahneman, 1992;
Wakker & Tversky, 1993; Schmidt & Zank, 2001; Safra
& Segal, 2001), is that outcome probabilities enter into
assessments of risky prospects or gambles through a
transformation that subjectively weights objective probabilities. The most common weighting effects are
inflation of small probabilities and devaluation of large
probabilities (Edwards, 1961; Tversky & Fox, 1995;
Tversky & Wakker, 1995; Wu & Gonzalez, 1996; Prelec,
1998; Gonzalez & Wu, 1999; Abdellaoui, 2000; Bleichrodt & Pinto, 2000; Luce, 2001) that give rise to the
familiar inverse S-shape of the probability weighting
Corresponding author. Fax: +1-617-812-6515.
E-mail address: [email protected] (D.E. Bell).
function. Several theories propose different weighting
functions for outcomes interpreted as gains and losses,
but we consider only a single function in this paper. We
often refer to it as a pwf, short for probability weighting
function.
Fig. 1 illustrates pwfs p that inflate small probabilities
½pðpÞ4p and deflate large probabilities ½pðpÞop:
Figs. 1a and b have pðpÞ ¼ p near p ¼ 0:375; the others
have pð12Þ ¼ 12: All but Fig. 1b satisfy an independence
axiom that is inconsistent with prevailing views on pwfs.
This axiom, referred to below as binary even-chance
independence, is the central topic of the paper. Despite
the fact that it departs substantially from prevailing
views, we will see that it allows weighting functions
which are similar to those proposed by others.
The EU assumption most responsible for behavioral
violations of EU is the independence axiom or substitution principle (Samuelson, 1952; Herstein & Milnor,
1953; Jensen, 1967). Our purpose here is to examine the
effects of a severely restricted independence axiom on
non-EU and the pwf. We do this in the context of RDU
(Quiggin, 1993), which is perhaps the most widely
analyzed descriptive alternative to EU. It will become
apparent that our restricted independence axiom is
antithetical to some views about RDU and is incompatible with conditions proposed by others for probability
weights. There is, however, no fundamental conflict
between the axiom and RDU, and, as suggested above,
0022-2496/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.
doi:10.1016/S0022-2496(02)00023-8
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
(a) 0
0.2
0.4
0.6
0.8
1
1
0
(b) 0
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
(c)
245
0
0.2
0.4
0.6
1
0.8
0
(d) 0
0:547
Fig. 1. (a) A pwf that satisfies BECI. (b) pðxÞ ¼ expððlnðxÞÞ
Þ for all 0oxo1: (c) p based on a w defined by a quintic on [0,0.5]. (d) p based on
a w defined by a linear plus exponential on [0,0.5] with c ¼ 8 (refer to Example 6).
we will argue that the axiom’s effects on RDU can be
practically indistinguishable from the effects of other
conditions that constrain probability transformations.
In this vein, the paper can be viewed as an attempt to see
how much of EU’s independence axiom can be retained
while preserving the integrity of the RDU viewpoint. At
the same time, our axiom accommodates but does not
require a variety of behavioral possibilities that are not
associated with the RDU model and are not usually
allowed by constraints proposed by others.
The restricted independence axiom we focus on says
that if x is a two-outcome gamble with certainty
equivalent c; then the even-chance lottery that yields x
or c; each with probability 12; is also indifferent to c: We
refer to this as binary even-chance independence, or
BECI. Our choice of the mixing probability 12 is
motivated by symmetry, both intuitional and analytical.
A fixed mixing probability other than 12 could be
considered, but we will not do so here. We shall see
later (Theorem 1) that if every mixing probability were
allowed for the indifference conclusion, then RDU
would reduce to EU. The following example conveys the
essence of BECI and issues it confronts.
Example 1. Paul has decided that he is indifferent
between $3000 with certainty and a binary gamble x that
pays $10,000 with probability 38 and nothing otherwise.
He is then asked to compare options A and B that
involve a fair coin toss.
A: Get $3000 regardless of whether heads or tails
occurs;
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
246
5
4
0.8
3
0.7
0.6
2
0.5
1
0.4
0
0.3
0.75
-1
0.8
0.85
0.9
0.2
-2
0.1
0
(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-3
(b)
Fig. 2. (a) w defined by straight line segments on [0,0.5] and extrapolated by Eq. (1). (b) The curve in (a) extrapolated to [0.75, 0.904].
B: Get $3000 if heads occurs and x if tails occurs.
Should Paul be indifferent between A and B? Will he in
fact be indifferent between them?
EU says that Paul should be indifferent between A
and B because this is implied by independence, even in
the restricted BECI version. But if Paul is unfamiliar
with EU, he may still be indifferent between A and B:
After all, if heads occurs, he gets $3000 in either case,
and if tails occurs, he is back to the original comparison
between $3000 and x:
A different impression could arise when outcome
probabilities are specified holistically:
A has pr. 1 for $3000;
5
B has pr. 16
for $0, pr. 12 for $3000, pr.
3
16
for $10,000.
Without the fair-coin dependency, it is far from obvious
that Paul will be indifferent between gambles A and B:
He may decide in fact that he definitely prefers one to
the other. This would violate BECI and EU, but is not
inconsistent with RDU.
Nevertheless, we will show that indifference between
A and B and general adherence to BECI can also be
consistent with RDU and with violations of EU that are
well known from Allais’s (1953, 1979) analyses and later
works.
We will show that BECI’s modest infusion of the
classical independence axiom into RDU can at least
partially preserve the integrity of RDU and allow pwfs
that are similar to—but not identical with—forms
proposed by others.
The next section outlines common versions of RDU
and EU and comments on independence axioms. We
assume for both models that the outcome set is a real
interval and that the utility function u for outcomes is
continuous and increasing over the interval. We denote
the pwf for RDU by p and assume that it is an
increasing and continuous map from ½0; 1 into ½0; 1
with pð0Þ ¼ 0 and pð1Þ ¼ 1: After the RDU and EU
models are introduced, we note that the generalization
of BECI to binary independence with biased coins
reduces RDU to EU.
Section 3 begins our analysis of BECI by noting in the
RDU context that it is equivalent to the functional
equation
p
p
½1 pðpÞ
2
1þp
¼ pðpÞ 1 p
for 0ppp1:
ð1Þ
2
The combination of RDU and BECI places no new
constraints on u and allows any p that satisfies (1) and is
an increasing onto map. We will see that (1) has
substantial mathematical implications. It constrains p in
interesting ways and can give rise to unrealistic pwfs (see
Fig. 2). On the other hand, it allows forms for p that
emulate effects observed by others that justify its serious
consideration in the RDU context.
Two consequences of (1) noted in Section 3 are called
conditions near 0 and near 1. The condition near 0 says
that pðpÞ=pð2pÞ approaches 1 pð12Þ as p-0: This
happens to be violated by most nonlinear forms for p
proposed in the literature (see Section 5). Another
consequence of (1) is that p cannot be concave near 0
and convex near 1 when pð12Þa12: Although this also goes
against prevailing wisdom about p; we argue that it
should not rule out BECI as a potential descriptive
condition.
The final part of Section 3 is based on the observation
that specification of p on an interval of length 12
determines it throughout ½0; 1 when (1) is presumed.
However, with p continuous and increasing on the
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
length-12 interval, its extension by (1) might fail to
increase throughout ½0; 1 and can even have singularities. Examples illustrate these phenomena. Because we
presume throughout that p is increasing and continuous
on ½0; 1; we use w in place of p when considering effects
of (1). For example, if w is increasing and continuous on
a length-12 interval in ½0; 1; and if (1) is assumed for w in
place of p throughout ½0; 1; then w might not be
increasing over ½0; 1:
Section 4 addresses the increasingness issues when w is
assumed to be continuous and increasing on ½0; 12 with
wð0Þ ¼ 0owð12Þo1: With an additional assumption of
differentiability on ½0; 12; we establish a sufficient
condition for increasingness over ½0; 1 based solely on
the behavior of w on ½0; 12: The condition is then used to
prove increasingness throughout ½0; 1 of some basic
analytical forms for w on ½0; 12 when w is extended to
½0; 1 by (1).
Section 5 demonstrates that p for (1) can have
approximately the inverse S-shape often attributed to
pwfs. We then consider six parametric forms for p
proposed by others. Three of the six violate (1), and the
other three are consistent with (1) if and only if pðpÞ ¼ p
for all p: Hence, if C is a set of conditions which implies
one of the six parametric forms then, in the RDU
context, either C and BECI are incompatible or the
combination of the two reduces RDU to EU.
The pwfs allowed by BECI that most closely resemble
forms proposed by others are continuous and increasing
but not fully differentiable. This is true of Fig. 1a, where
the left and right derivatives of p are not always equal:
see Section 5. Fig. 1b pictures a differentiable pwf that
does not satisfy BECI. The reason that BECI approximations of such traditional transformations are not
fully differentiable is that differentiable pwfs that satisfy
BECI usually require pð12Þ ¼ 12: Examples are shown in
Figs. 1c and d. We elaborate on these and on
differentiability in Section 6, where it is shown (under
stated conditions) that if w satisfies (1) and is differentiable at 12 then it is differentiable everywhere. A simple
example (not discussed later) is p defined as 12p þ p2 on
½0; 12 and by (1) for ð12; 1: This function is S-shaped,
continuous and differentiable throughout ½0; 1:
2. Preference between gambles
Let X be the set of finite-support probability
distributions or gambles x; y; y on a nondegenerate
real interval I of outcomes a; b; y: The convex
combination lx þ ð1 lÞy of x; yAX with 0plp1 is
the gamble with probability lxðaÞ þ ð1 lÞyðaÞ for
aAI: We intend that the convex combination lx þ ð1 lÞy is a gamble entirely defined by its probability
distribution. The related compound gamble, x with
247
probability l and y with probability 1 l; which we do
not consider, may or may not be evaluated equivalently.
Let h be a preference-or-indifference relation on X
with indifference B and strict preference g defined by
xBy
if xhy and yhx;
xgy
if xhy and not ðyhxÞ:
Let x!y mean that ygx: We assume that preference
increases over I so that a!b when a; bAI and aob:
As mentioned above, we denote a pwf by p and
assume that p is strictly increasing and continuous from
½0; 1 onto ½0; 1; so pð0Þ ¼ 0 and pð1Þ ¼ 1: If pðpÞ ¼ p
for all pA½0; 1 then p is the identity. We assume also for
the RDU model and its EU specialization that the utility
function u on outcomes is continuous and strictly
increasing from I into R:
Let Xn denote the subset of gambles that have positive
probabilities for exactly n points in I; so
Xn ¼ fxAX : jfaAI : xðaÞ40gj ¼ ng:
We identify X1 with I and refer to members of X2 as
binary gambles.
Suppose xAXn has positive probability pi ¼ xðai Þ for
outcome ai with a1 oa2 o?oan : The RDU utility for x
with respect to p and u is
"
!
!#
n1
n
n
X
X
X
p
pi p
pi uðaj Þ
Epu ðxÞ ¼
j¼1
i¼j
i¼jþ1
þ pðpn Þuðan Þ:
When p is the identity, this simplifies to the EU utility
n
X
Eu ðxÞ ¼
pj uðaj Þ:
j¼1
The RDU representation for h with respect to p and u
is
xhy3Epu ðxÞXEpu ðyÞ;
for all x; yAX :
The corresponding EU representation is
xhy3Eu ðxÞXEu ðyÞ;
for all x; yAX :
More general treatments of these representations, which
allow arbitrary outcome sets and do not presume
continuity, are available in Fishburn (1982) for EU
and Quiggin (1993) and Abdellaoui (2002) for RDU.
Quiggin (1993) provides extensive analyses of axioms
and economic applications for RDU.
As background for BECI, we note four independence
axioms that hold for the EU representation but are
violated by most RDU proposals.
IA1. For all x; y; zAX and all 0olo1 : xgy ) lx þ
ð1 lÞzgly þ ð1 lÞz:
IA2. For all x; y; zAX : xBy ) 12x þ 12zB12y þ 12z:
IA3. For all xAX2 ; all aAI and all 0olo1: xBa )
lx þ ð1 lÞaBa:
BECI. For all xAX2 and all aAI: xBa ) 12x þ 12aBa:
248
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
Despite arguments in favor of the axioms, which
include two-stage options (see Example 1) with fair ðl ¼
1
1
2Þ or biased ðla2Þ coins and avoidance of money pumps
(Raiffa, 1968), extensive empirical analyses have shown
that they are often violated in practice. IA1 is Jensen’s
(1967) axiom, and IA2 is due to Herstein and Milnor
(1953). In conjunction with ordering and Archimedean
axioms, each implies the existence of u on I that satisfies
the EU representation.
BECI is a weak version of the Herstein–Milnor
axiom, and IA3 is a biased-coin generalization of BECI.
Although we suggest that BECI could have descriptive
validity and will argue for its compatibility with RDU,
we will not do this for IA3. There are two reasons. The
first is that Allais-type examples severely challenge the
descriptive robustness of IA3.
viors of p near 0 and 1. By setting p ¼ 2t and then p ¼
1 2t in (1), we get
1
pðtÞ
1
for 0oto ;
ð2Þ
p þ t ¼ 1 þ pðtÞ 2
pð2tÞ
2
1
1 pð1 tÞ
1
p t ¼ pð1 2tÞ
for 0oto : ð3Þ
2
1 pð1 2tÞ
2
Let
a¼p
1
:
2
Then, under continuity, (2) and (3) give
pðtÞ
¼ 1 a;
the condition near 0: lim
t-0 pð2tÞ
the condition near 1:
Example 2. Paula is indifferent between $2,000,000 with
certainty and the binary gamble x that pays $3,000,000
with probability 0.98 and nothing otherwise. IA3 says
that she will be indifferent between $2,000,000 with
certainty and ð0:005Þx þ ð0:995Þa; where a is a sure thing
for $2,000,000. However, the convex combination has
pr: 0:0001 for $0; pr: 0:995 for $2; 000; 000;
pr: 0:0049 for $3; 000; 000;
and Paula says she prefers this to $2,000,000 because
there is only one chance in 10,000 that it returns $0 and
49 chances in 10,000 that it returns $1,000,000 more than
the expected outcome of $2,000,000.
The second reason is that IA3 reduces RDU to EU.
Theorem 1. If IA3 holds for the RDU representation,
then p is the identity.
The proofs of Theorem 1 and subsequent results are
in the appendix.
3. BECI basics
Our fundamental result for BECI is
Theorem 2. BECI holds for the RDU representation if
and only if (1) holds.
We show next that the strict preference companion of
BECI follows from it and the RDU representation.
Lemma 1. Suppose BECI and the RDU representation
hold. If xAX2 and bAI then xgb ) 12x þ 12bgb; and
bgx ) bg12x þ 12b:
Although (1) allows a variety of shapes for p; it
implies special conditions between pð12Þ and the beha-
lim
t-0
1 pð1 tÞ
¼ a:
1 pð1 2tÞ
For example, given d40; there exists an e40 such that
1 þ pðtÞ pð12 þ tÞA½1 a d; 1 a þ d for all toe; so
pðtÞ=pð2tÞA½1 a d; 1 a þ d for all toe: It follows
by taking d towards 0 that the limit defined in the
condition near 0 exists and equals 1 a: Both limits are
obviously in ð0; 1Þ and, because of their dual natures,
their sum equals 1.
When p is near 0, the condition near 0 says that pðp2Þ is
approximately ð1 aÞpðpÞ; when p is near 1, the
condition near 1 says that 1 pð1þp
2 Þ is approximately
a½1 pðpÞ: We say that p is concave fconvexg near 0 if
for some e40; pðp2Þ412pðpÞfpðp2Þo12pðpÞg for all 0opoe:
Similarly, p is concave fconvexg near 1 if for some e40;
1þp
1
1
1 pð1þp
2 Þo2½1 pðpÞf1 pð 2 Þ42½1 pðpÞg for all
1 eopo1:
Most studies of pwfs conclude that ao12 and suggest
that pðpÞ ¼ p for p in the neighborhood of 0.3–0.4.
Many, including Wu and Gonzalez (1996, 1999), Prelec
(1998), Abdellaoui (2000) and Bleichrodt and Pinto
(2000), also support the hypothesis that p begins with a
strictly concave segment ½pðlx þ ð1 lÞyÞ4lpðxÞ þ
ð1 lÞpðyÞ; 0olo1; xay followed by a strictly
convex segment ½pðlx þ ð1 lÞyÞolpðxÞ þ ð1 lÞpðyÞ;
0olo1; xay; hence that p is concave near 0 and
convex near 1. The latter behavior is impossible under
BECI in the RDU context unless a ¼ 12:
Lemma 2. Suppose BECI and the RDU representation
hold. If pð12Þo12 then p is concave near 0 and concave near
1. If pð12Þ412 then p is convex near 0 and convex near 1.
Lest one conclude that this dooms BECI as a viable
descriptive condition, we emphasize that the conditions
near 0 and 1 apply only to p’s extremes and do not
invalidate the phenomena of subjective overvaluation of
small probabilities and subjective undervaluation of
large probabilities. For example, if a is near 0.4 and the
ratios in the conditions are near their limit values within
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
249
0.01 of the extremes, then p near 0 and 1 may look like
the parts of curve 1 in Fig. 1 near 0 and 1. In general, the
intervals that contain 0 and 1 and over which p’s
curvature does not reverse can be arbitrarily short.
An important consequence of BECI’s even-chance
feature is that p’s arguments in (1), namely p=2; p and
ð1 þ pÞ=2; span exactly half of ½0; 1: If p is specified on a
length-12 interval in ½0; 1 then (1) effectively determines p
on all of ½0; 1: We comment on the construction of p for
such a specification in the next several paragraphs, then
conclude the section with cautionary examples and a
final comment on extension by (1).
Suppose first that p is specified on ½0; 12,f1g with p
continuous and increasing on ½0; 12; pð0Þ ¼ 0 and pð1Þ ¼
1: We assume that p satisfies the condition near 0.
Because 2tp123tp14; we first extend p to ð12o34; and p
thus extended is continuous at 12 because it satisfies the
condition near 0. The next step extends p to ð34; 78 using
(2) for 14otp38 because 2tp343tp38: Each succeeding
step covers half the interval between the right end of the
interval thus far covered and 1.
The procedure is similar but two-sided if p is first
specified on ½l; l þ 12 with 0olo12: An extension step in
either direction covers half the remaining interval in that
direction. Upward extension can use (2), downward
extension (3), or modified expressions equivalent to (1).
Our next lemma shows that the upward extension
from ½0; 12 satisfies some of the properties presumed
earlier for p: However, it need not satisfy all such
properties, including monotonicity. As explained earlier,
we therefore use another designation, w; instead of p:
The oddity of Example 3 is compounded in the next
example. For convenience there, w is defined to be
constant over subintervals of ½0; 12: This violates strict
monotonicity on ½0; 12; but essentially the same results
will be obtained if the constant pieces are modified by
giving them small positive slopes.
Lemma 3. Suppose w : ½0; 1-R,fN; Ng is increasing and continuous on ½0; 12; has 0 ¼ wð0Þowð12Þ ¼
aowð1Þ ¼ 1; and satisfies (1) in place of p: Then w
increases on fð2k 1Þ=2k : k ¼ 1; 2; y; g and
Although we have emphasized extension of w to ½0; 1
from a length-12 interval, complete extension can proceed
from specification of w on disjoint intervals with total
length slightly greater than 14 provided that certain
precautions are taken. Suppose in particular that w is
specified on ½e; 2e,½12; 34 with small e40; and consider
(1) for w in the form
k
2 1
lim w
¼ 1:
k-N
2k
Unfortunately, the nice behavior of w on fð2k 1Þ=2k g by extension through (1) from w on ½0; 12 does not
necessarily extend to other properties of p:
Example 3. Suppose wðpÞ ¼ 3p2 for 0ppp12; where the
constant 3 is chosen to satisfy the condition near zero.
Using (2), we have
wðpÞ ¼ 32ð1 2p þ 2p2 Þ
for 12ppp34;
so wð0:75Þ ¼ 15
16 ¼ 0:9375: But, by (2) for w;
wð0:3Þ
wð0:8Þ ¼ 1 þ wð0:3Þ wð0:6Þ
0:27
¼ 1 þ 0:27 ¼ 0:9238;
0:78
so w decreases over parts of ½0; 1:
Example 4. Let w on ½0; 12 be the continuous four-piece
function defined by
2p
1
4
for 0ppp18;
for 18ppp14;
5
;
4p 34 for 14ppp16
1
2
for
5
1
16ppp2:
When w is extended by (2), we obtain the picture
shown in Fig. 2 for wðpÞ from p ¼ 0 to p ¼ 0:904:
Fig. 2a shows a decrease in w near 0.6. In the extension
of Fig. 2b, w goes negative near a¼0:80646
to a local
’
1
minimum of wð13
Þ
¼
and
remains
negative
until
16
2
b¼0:81392:
Because w ¼ 0 at a and b; singularities
’
0
0
0
occur in the ð78; 15
16 region at a and b ; where a ¼
0
ð1 þ aÞ=2 and b ¼ ð1 þ bÞ=2: Other singularities
occur whenever w crosses the abscissa. If we define the
value of w at a singularity by w’s ‘limit’ as the singularity
is approached from the left, then wða0 Þ ¼ N and
wðb0 Þ ¼ þN:
wð2tÞ ¼
wðtÞ
:
1 þ wðtÞ wð12 þ tÞ
When t ¼ e; this constrains the values of wð2eÞ; wðeÞ and
wð12 þ eÞ: Assuming consistency at these values, the
formula extends w to ½2e; 4e; then to ½4e; 8e; y; and
also to ½e=2; e; then to ½e=4; e=2; y . Continuity would
require wðtÞ-0 as t-0 for extension downward, and
consistency near 12 for extension upward. Once w is
determined on ½0; 34; its upward extension to ½34; 78;
½78; 15
16; y would proceed as before.
Because of alignment/consistency problems in extensions based on intervals of length less than 12; we focus
henceforth on extensions based on ½0; 12 with an
exception noted in Section 5 that is based on
½0:125; 0:625:
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
250
These functions and their extensions to ½12; 1 are shown
in Fig. 3.
Each of (4a) through (4b) satisfies the condition near
0 and has wð0Þ ¼ 0: The first has wð12Þ ¼ 1 21=2 ¼0:29289
and the others have wð12Þ ¼ 12: Although
’
their full extensions to ½0; 1 are not typical of shapes for
p in the literature, we will remedy this in the next
section.
Our sufficient condition is set within the following
context.
4. Increasing extensions
Although Examples 3 and 4 show that extension of w
by (1) from its specification on ½0; 12 can grossly violate
properties presumed earlier for p; such violations need
not occur. In this section, we establish a sufficient
condition under which extension by (1) from an
increasing and continuous w on ½0; 12 is increasing and
continuous on ½0; 1: The condition is then illustrated for
the following four w’s on ½0; 12:
pffiffiffi
pffiffiffi
wðpÞ ¼ ð 2 1Þ p;
ð4aÞ
wðpÞ ¼ 0:37p þ 0:8p2 þ 0:92p3 ;
ð4bÞ
wðpÞ ¼ e1=2 pep
ðe1=2 ¼0:60653Þ;
’
ð4cÞ
wðpÞ ¼ e1=2 pep
ðe1=2 ¼1:64872Þ:
’
ð4dÞ
Axiom 1. w on ½0; 12 into R is continuous, increasing,
satisfies the condition near 0 and has wð0Þ ¼ 0 and
wð12Þo1: In addition, w is differentiable on ð0; 12Þ:
The condition is based solely on the behavior of w and
w0 on ð0; 12Þ: Let w0 ðf ðtÞÞ ¼ dwðf ðtÞÞ=dt ¼ ½dwðf ðtÞÞ=
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
(a) 0
0.2
0.4
0.6
0.8
1
0
(b) 0
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
(c) 0
0.2
0:5
0.4
0:5
Fig. 3. (a) wðxÞ ¼ ð2 1Þ x
x expð0:5 xÞ for xp0:5:
0.6
0.8
0
(d) 0
1
2
3
for xp0:5: (b) wðxÞ ¼ 0:37x þ 0:8x þ 0:92x for xp0:5: (c) wðxÞ ¼ x expðx 0:5Þ for xp0:5: (d) wðxÞ ¼
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
df ðtÞ½df ðtÞ=dt; and let
GðtÞ ¼
wðtÞw0 ð2tÞ w0 ðtÞwð2tÞ½1 wð2tÞ
;
wðtÞ½1 wð2tÞfwð2tÞ wðtÞ½1 wð2tÞg
1
0oto :
4
Axiom 1 implies that 14wð2tÞ4wðtÞ4wðtÞ½1 wð2tÞ;
so the denominator of G is positive when the axiom
holds. Also let
2t þ 2k1 1
ak ðtÞ ¼
2k
1
for 0ptp ; k ¼ 1; 2; y :
4
Thus, a1 ðtÞ ¼ t; a2 ðtÞ ¼ ð2t þ 1Þ=4; y; and
a1 ð14Þ ¼ a2 ð0Þoa2 ð14Þ ¼ a3 ð0Þoa3 ð14Þ
¼ a4 ð0Þo?o12
with
2akþ1 ðtÞ ¼ 12 þ ak ðtÞ for k ¼ 1; 2; y :
Finally, let
sk ðtÞ ¼
w0 ðak ðtÞÞ
1
for 0oto ; k ¼ 1; 2; y :
wðak ðtÞÞ
4
Because 0oak ðtÞo12 and w is increasing on ½0; 12 when
Axiom 1 holds, the axiom implies that sk ðtÞ40 in all
cases.
Theorem 3. Suppose w : ½0; 1-R satisfies Axiom 1. If
N
X
1
ð5Þ
sk ðtÞoGðtÞ for all 0oto ;
4
k¼2
and if w satisfies (1) in place of p; then w is continuous and
increasing on ½0; 1Þ; and limp-1 wðpÞ ¼ 1:
Thus, when Axiom 1 and (5) hold, the continuous
extension of w from ½0; 12 to ½0; 1 via (1) has the
properties assumed initially for p and satisfies BECI in
the context of the RDU representation. Although
Example 3 and a slight modification of Example 4 show
that (5) can easily fail when Axiom 1 holds, many
‘reasonable’ w that satisfy Axiom 1 do satisfy (5).
The hypotheses of Theorem 3 imply that w has left
and right derivatives at 12; 34; 78; y; but they do not imply
that w is differentiable there. This is evident in Fig. 3,
where the extensions by (1) of (4a)–(4d) have kinks at 12;
3
4; y: The extensions apparently increase over ½0; 1; but
Theorem 3 allows rigorous proofs. We conclude this
section with a summary for
pffiffiffieach case.
pffiffiffi
(4a): wðpÞ ¼ c p; c ¼ 2 1; on ½0; 12: Differentiation and simplification give
1
1
GðtÞ ¼ pffiffi
;
¼ pffiffi
t½1 c2 2t
t½1 0:34315t
sk ðtÞ ¼
1
2t þ 2k1 1
for kX2:
251
Within the range of 0 to 14 for t; we have
N
N
X
X
1 1
max
sk ðtÞ ¼
sk ð0Þ ¼ 1 þ þ þ ?o1
t
3 7
2
k¼2
1 1
þ þ?¼2
2 4 1
o 2:1877 ¼ G
¼ min GðtÞ:
t
4
þ
Hence (5) holds and, by Theorem 3, w extends by (1) to
a continuous and increasing function on ½0; 1:
(4b): wðpÞ ¼ ap þ bp2 þ cp3 ; ða; b; cÞ ¼ ð0:37; 0:80;
0:92Þ; on ½0; 12: Evaluation of GðtÞ shows that it is
minimized near t ¼ 0:05 and that GðtÞ45:6 for all
0oto14: We have
!
1
a þ 2bak ðtÞ þ 3cak ðtÞ2
sk ðtÞ ¼ k1
:
2
aak ðtÞ þ bak ðtÞ2 þ cak ðtÞ3
This decreases in t; so max sk ðtÞ ¼ sk ð0Þ: Computations
give s2 ð0Þ þ s3 ð0Þ þ s4 ð0Þo4:7: For kX5; sk ð0Þo
4=2k1 ¼ 1=2k3 ; so s5 ð0Þ þ s6 ð0Þ þ ?o12: Therefore
(5) holds:
N
X
max
sk ðtÞo5:2o5:6omin GðtÞ:
k¼2
(4c): wðpÞ ¼ cpep ; c ¼ e1=2 ; on ½0; 12: Here
GðtÞ ¼
2et ½1 þ 2cð1 þ tÞe2t :
½1 2cte2t ½2et 1 þ 2cte2t This is minimized at t ¼ 0; where Gð0Þ ¼ 4:4261:
Moreover,
1
2k
sk ðtÞ ¼ k1 1 þ
;
2
2t þ 2k1 1
so max sk ðtÞ ¼ sk ð0Þ: We have
N
X
1 1 1
max
sk ðtÞ ¼ 1 þ 2 1 þ þ þ þ ?
3 7 15
k¼2
1 1 1
o1 þ 2 1 þ þ þ
3 7 15
1
1
þ þ?
þ 2
16 32
¼ 4:3357o4:4261 ¼ min GðtÞ;
so (5) holds.
(4d): wðpÞ ¼ cpep ; c ¼ e1=2 ; on ½0; 12: Here
4cð1 tÞe2t 2
;
ð1 2cte2t Þð2 et þ 2cet Þ
1 2k1 2t þ 1
sk ðtÞ ¼ k1 k1
k ¼ 2; 3; y :
2
2
þ 2t 1
GðtÞ ¼
GðtÞ P
decreases from about 4.59 at t ¼ 0 to 1.56 at t ¼ 14;
and N
2 sk ðtÞ decreases from
PN3.72 at t ¼ 0 to1 1.43 at
t ¼ 14: GðtÞ remains above
2 sk ðtÞ over ½0; 4; so (5)
holds.
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
252
5. Comparisons
6. Smooth BECI transformations
In this section, we note that (1) accommodates pwfs
that are quite similar to pwfs proposed by others. We
then prove that essentially none of those functions
satisfies (1) or BECI unless it is the identity.
Fig. 1a shows a pwf that satisfies (1) approximately by
a series of curve-fitting adjustments. Our fit of the curve
to (1) fixed pðpÞ ¼ p for p ¼ 0:375 and worked outward
from ½0:125; 0:625: The curve of Fig. 1a is not perfectly
smooth (cf. Fig. 3) but is continuous and increasing. We
constructed it to minimize differentiability kinks and
simulate pwfs that appear in recent studies. It is concave
near 1 (cf. Lemma 2) but otherwise resembles the
smooth concave–convex standards, an example of which
is Fig. 1b.
We consider six such standard forms for p on ½0; 1:
We conclude our technical analyses with observations
on differentiability of pwfs that satisfy BECI in the
RDU context. We focus first on differentiability at 12 and
observe later that this implies differentiability at other
key extension points, namely 34; 78; y: Our initial result at
1
2 is based on the following strengthening of Axiom 1.
a
pðpÞ ¼ ebðln pÞ ;
pðpÞ ¼
pðpÞ ¼
p
0oao1; b40;
g
½pg
þ ð1 pÞg 1=g
apg
apg
;
þ ð1 pÞg
a
pðpÞ ¼ eðb=aÞð1p Þ ;
pðpÞ ¼ pb ;
ð6aÞ
;
g40;
ð6bÞ
a; g40;
ð6cÞ
aa0; b40;
ð6eÞ
b40;
pðpÞ ¼ ð1 a ln pÞb ;
ð6dÞ
a40; b40:
ð6fÞ
All six are included in Prelec (1998). Prelec (1998) and
Luce (2001) provide axioms for (6a). Earlier references
include Tversky and Kahneman (1992) for (6b) and
Tversky and Fox (1994) for (6c).
When p is not the identity, none of the six satisfies (1),
so all violate BECI for the RDU representation.
Moreover, all but (6c) violate the condition near 0
unless p is the identity, and (6c) reduces to the identity if
it satisfies both the condition near 0 and the condition
near 1.
Theorem 4. The only cases of (6a)–(6f) that satisfy the
condition near 0 are (6b) and (6c) with pðpÞ ¼ p for all
pA½0; 1; and (6c) with a ¼ 2g 1: If (6c) satisfies the
condition near 0 and the condition near 1; then p is the
identity.
It follows from differentiability results in the next
section that (6a)–(6f) cannot satisfy BECI except in
special cases. Our proof of this in the appendix focuses
on the more general condition near 0.
An obvious corollary of Theorem 4 is that if C is a set
of conditions that implies one of (6a)–(6f) within the
RDU representation, then either C and BECI are
incompatible or they jointly reduce RDU to EU.
Axiom 2. Axiom 1 holds, w satisfies (1) and is increasing
on ½0; 1; and w is twice differentiable near 0 with w0 ð0Þ
and w00 ð0Þ finite and w0 ð0Þ40:
Axiom 2 does not apply to certain forms of w like
(4.1) that do not have finite derivatives at 0. We denote
the limit of w0 ðtÞ as t approaches 12 from below (from
þ
above) by w0 ð12 Þ½w0 ð12 Þ:
Lemma 4. Suppose w satisfies Axiom 2. Then w is
differentiable at 12 if and only if
00
1
3
1 w ð0Þ
:
ð7Þ
¼ w0 ð0Þ þ w
w0
2
4
2 w0 ð0Þ
We continue by involving the third derivative near 0
to conclude that wð12Þ ¼ 12 when w00 ð0Þa0: This conclusion does not presume differentiability at 12:
Lemma 5. Suppose w satisfies Axiom 2 and has a third
derivative near 0 with w000 ð0Þ finite. Then
1
1 00
2 w 2 w ð0Þ ¼ 0:
The techniques of the proofs of Lemmas 4 and 5 can
be extended to determine conditions for higher-order
derivatives at 12: We outline only the second-order result.
Lemma 6. Suppose Axiom 2 holds, w is twice differentiable on ð0; 12Þ; is differentiable at 12; and has third and
fourth derivatives near 0 with w000 ð0Þ and w0000 ð0Þ finite.
Then w is second differentiable at 12 if and only if
4 1 1 w000 ð0Þ
00 1
00
w ð2 Þ ¼ w ð0Þ þ w 2 3
6 w0 ð0Þ
00 2
w ð0Þ
3
2w 12
:
2
w0 ð0Þ
When wð12Þ ¼ 12; which is true by Lemma 5 when
w ð0Þa0; the preceding equality simplifies to
1
1 w000 ð0Þ 1 w00 ð0Þ 2
w00
:
¼ w00 ð0Þ þ
2
2 w0 ð0Þ 2 w0 ð0Þ
00
Under Axiom 2, it is easily seen that w is differentiable
under extension from ½0; 12 to ½0; 1 within the open
intervals ð12; 34Þ; ð34; 78Þ; y: Hence, the question of differentiability everywhere concerns only differentiability at
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
1 3 7
2; 4; 8; y
. Lemma 4 addressed differentiability at 12: Our
next result shows that differentiability at 12 implies
differentiability at the other points in question.
253
d ¼ 8ð1 aÞ 4b 2c:
ground, the present paper might be viewed as an attempt
to see how much of EU’s independence axiom can be
retained without destroying the integrity of the RDU
viewpoint and the behaviors it represents. Our contention is that BECI accomplishes these ends.
A subtheme of the paper is that several roads lead
from RDU to EU, including independence axioms
stronger than BECI, and BECI in conjunction with
strict concavity–convexity of the pwf p or other
conditions that lead to parametric forms for p: Another
route from RDU to EU is a contextual uncertainty
condition in Bell and Fishburn (2000). But BECI by
itself does not have this effect.
The major technical focus of the paper is the
functional equation (1) that is equivalent to BECI in
the RDU setting. We have seen that (1) accommodates a
wide range of shapes for p but can lead to unusual and
even bizarre possibilities when applied under extension
to ½0; 1 from specification on a length 12 interval in
½0; 1 such as ½0; 12: Nevertheless it can, with careful
assessment and to a reasonable approximation, emulate
weighting functions that have emerged as promising
descriptors of probability distortion.
Is BECI a viable descriptive condition? We have
suggested that it may be, but remain open on the point.
In any event, BECI is empirically testable, and we look
forward to assessment of its descriptive tenability.
When 0oao1; w is S-shaped; when 1oao1:5; w is
inverse S-shaped.
Appendix
Theorem 5. Suppose Axiom 2 holds and w is differentiable at 12: Then w is differentiable everywhere.
The proof method for Theorem 5 extends easily to all
degrees of differentiability. For example, if w is assumed
to be twice differentiable on ½0; 12 and at 12; then it is twice
differentiable at all bk and is therefore twice differentiable everywhere.
Figs. 1c and d are examples of differentiable w
functions that satisfy BECI. Fig. 1c takes w as a quintic
polynomial on ½0; 12; and Fig. 1d has a linear plus
exponential form on ½0; 12: We discuss the latter in
Example 6, preceded by comments on quartic polynomials in Example 5.
Example 5. Suppose w satisfies (1) with wðpÞ ¼ ap þ
bp2 þ cp3 þ dp4 on ½0; 12: Then w is continuous and
differentiable on ½0; 1 if and only if the coefficients satisfy
b ¼ 12fð3 þ 9aÞ þ ½ð3 þ 9aÞ2 þ 48að1 aÞða þ 2Þ1=2 g;
c ¼ 4b 2b=a þ 16ð1 aÞ;
Example 6. Suppose w satisfies (1) with wðpÞ ¼ ap þ
bð1 ecp Þ on ½0; 12: Then w is continuous and twice
differentiable if and only if
b ¼ ð1 cec=2 =4Þ=½2ð1 ec=2 Þ c;
Proof of Theorem 1. We assume that IA3 holds for the
RDU representation. With aoc in I; assume without
loss of generality that uðaÞ ¼ 0 and uðcÞ ¼ 1: Let xAX2
have probabilities 1 p for a and p for c: Then
Epu ðxÞ ¼ pðpÞ:
a ¼ 1 2bð1 e
c=2
Þ:
The cases of 5pcp10 are well-behaved. Moreover, w
is S-shaped when co3:18 and inverse S-shaped when
c43:18: The curve for c ¼ 8 is graphed in Fig. 1d.
7. Discussion
The RDU model in this paper, which follows
Quiggin’s (1993) formulation, is a generalization of
EU that accounts for subjective weighting or distortion
of probabilities in evaluation of gambles. Although
often presented as a descriptive theory, RDU has
normative overtones such as complete ordering and
reduction of compound lotteries to holistic gambles.
The main normative principle eschewed by RDU is
the independence axiom of EU that rules out the
probability-distortion phenomenon. Against this back-
Continuity assures that uðbÞ ¼ pðpÞ for some aoboc;
and with xBb; IA3 gives pðpÞ ¼ uðbÞ ¼ Epu ðlx þ ð1 lÞbÞ ¼ ½pðlp þ 1 lÞ pðlpÞuðbÞ þ pðlpÞ: Letting t ¼
lp þ 1 l; the equality for IA3 can be written as
p
p ð1 tÞ
1p
pðpÞ
; 0opptp1; po1:
¼ ½1 pðtÞ
1 pðpÞ
For any fixed b41 and probabilities p; q; s and t such
that pps; ppt and qpminfs; t=bg ¼ t=b; set
p
p
q ¼ ð1 sÞ
and bq ¼ ð1 tÞ
:
1p
1p
The preceding equality implies, after cancellation of the
pðpÞ=½1 pðpÞ terms, that
pðqÞ
1 pðsÞ
¼
:
pðbqÞ 1 pðtÞ
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
254
By varying p and taking q0 ¼ ð1 sÞp0 =ð1 p0 Þ and
bq0 ¼ ð1 tÞp0 =ð1 p0 Þ; we obtain pðq0 Þ=pðbq0 Þ ¼ ½1 pðsÞ=½1 pðtÞ: Because t can be arbitrarily close to 1
here, we have
0
pðqÞ
pðq Þ
¼
for all positive q; q0 o1=b:
pðbqÞ pðbq0 Þ
Equality also holds by continuity for q ¼ 1=b; so
pðqÞ
1
¼p
pðbqÞ
b
for all 0oqp1=b:
Now take q ¼ 1=b2 ; 1=b3 ; y to obtain
n
1
1
p n ¼ p
b
b
for n ¼ 1; 2; y :
Let a ¼ pð12Þ: When 1obo2 and bn ¼ 2m for positive
integers mon; we have
k ( 1=n )k
1
1
1
p km=n ¼ p m=n
¼
p m
¼ akm=n :
2
2
2
Because fkm=n : 1pmon; k ¼ 1; 2; yg is dense in ½0; 12;
it follows from continuity that
1
p w ¼ aw for all wX1:
2
Equivalently,
pðpÞ ¼ alog2 ð1=pÞ
for all 0ppp12:
Given 0opptp12; substitution in the equation that
concludes the opening paragraph and rearrangement
give
1
alog 1t
1
1 alog t
¼
a
1
log 1p
1a
1
log p
;
all logs base 2:
Take t ¼ 12 and p ¼ 14 in this equation to get
log 3
1
1
:
1þ ¼
a
a
This holds when a ¼ 12 and for no other a40: for wX0;
derivatives show that 1 þ w intersects wlog 3 only at w ¼
2: We conclude that
pðpÞ ¼ p
for all
0ppp1=2:
Eq. (1) shows that this extends uniquely under BECI to
the identity. &
Proof of Theorem 2. Assume that the RDU representation holds. Suppose BECI also holds, aoboc are
outcomes in I; xAX2 has probabilities 1 p for a and p
for c; and xBb: Then Epu ðxÞ ¼ uðbÞ; i.e.,
uðbÞ ¼ uðcÞpðpÞ þ uðaÞ½1 pðpÞ:
By BECI, 12x þ 12bBb and therefore Epu ð12x þ 12bÞ ¼ uðbÞ;
i.e.,
p
p
1þp
uðcÞp
þ uðbÞ p
p
2
2
2
1þp
þ uðaÞ 1 p
¼ uðbÞ:
2
Substitution here for uðbÞ from the preceding equation
and rearrangement give
½uðcÞ uðaÞ
p
1þp
p
½1 pðpÞ ¼ 0:
pðpÞ 1 p
2
2
Because uðcÞ uðaÞ40; the expression in braces equals
0 and gives the equation of (1) for 0opo1: Under the
original suppositions, including xBb; the derivation’s
steps are reversible. Since (1) holds trivially for pAf0; 1g;
the proof is complete. &
Proof of Lemma 1. Suppose x has probabilities 1 p for
a and p for c with aoc in I: Clearly, xgb ) boc:
Suppose xgb and apboc: With
p
p
1
1
1þp
Epu x þ b ¼ uðcÞp
þ uðbÞ p
p
2
2
2
2
2
1þp
þ uðaÞ 1 p
;
2
we use (1) to obtain
1
1
x þ bgb
2
2 1
1
3Epu x þ b 4uðbÞ
2
2
p
1þp
3½uðaÞ uðbÞ 1 p
4½uðbÞ uðcÞp
2
2
3½uðaÞ uðbÞ½1 pðpÞ4½uðbÞ uðcÞpðpÞ
3Epu ðxÞ4b
3xgb:
If xgb and boaoc; then clearly Epu ð12x þ 12bÞ4uðbÞ; so
þ 12bgb: The proof for bgx is similar. &
1
2x
Proof of Lemma 2. Given the initial hypotheses,
suppose a ¼ pð12Þo12: Let b be such that 1 a4b412:
By the condition near 0, pðpÞ=pð2pÞ4b for all sufficiently small p40; so p is concave near 0. By the
condition near 1, ½1 pð1 pÞ=½1 pð1 2pÞo1 b
for all sufficiently small p40; so p is concave near 1. The
proofs for a412 are similar. &
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
Proof of Lemma 3. Given the lemma’s hypotheses, let
ak ¼
2k1 1
;
2k
k ¼ 2; 3; y;
rk ¼ wða2 Þwða3 Þ?wðak Þ;
1
wðakþ1 Þ½1 wð2akþ1 Þ
w þ akþ1 ¼ 1 2
wð2akþ1 Þ
wðakþ1 Þ½1 wð12 þ ak Þ
:
¼1 wð12 þ ak Þ
It follows that
Because wðakþ1 Þoa; and aowð12 þ ak Þ by previous steps,
it follows that wð12 þ ak Þowð12 þ akþ1 Þ; so w increases on
fð2k 1Þ=2k : k ¼ 1; 2; yg: &
Proof of Theorem 3. Assume that w : ½0; 1-R satisfies
Axiom 1, (1) and (5). By (2) for w;
k ¼ 2; 3; y :
Note that 2akþ1 ¼ 12 þ ak ¼ ð2k 1Þ=2k : By (2),
wð12Þ
wðtÞ½1 wð2tÞ
;
w 12 þ t ¼ 1 wð2tÞ
w0
¼ a;
wð1Þ½1 wð12Þ
3
ð1 aÞr2
;
¼1
w
¼1 4
1
4
a
wð2Þ
wð3Þ½1 wð34Þ
7
ð1 aÞr3
w
¼1
;
¼1 8
3
8
a ð1 aÞr2
wð4Þ
and, by continuation,
k
2 1
ð1 aÞrk
w
;
¼1
k
2
a ð1 aÞðr2 þ r3 þ ? þ rk1 Þ
1
wðtÞw0 ð2tÞ w0 ðtÞwð2tÞ½1 wð2tÞ
þ
t
¼
;
2
½wð2tÞ2
wða2 ðtÞÞ½1 wð12 þ tÞ
w 12 þ a2 ðtÞ ¼ 1 ;
wð12 þ tÞ
¼
Because w increases on ½0; 12; we have 0ork oak1 ; so
rk -0 as k-N: In addition
1
0ptp :
4
wða2 ðtÞÞw0 ð12 þ tÞ w0 ða2 ðtÞÞwð12 þ tÞ½1 wð12 þ tÞ
½wð12 þ tÞ2
wða2 ðtÞÞ½1 wð12 þ tÞ
wð12 þ tÞ
(
)
wðtÞw0 ð2tÞ w0 ðtÞwð2tÞ½1 wð2tÞ
s2 ðtÞ
wð12 þ tÞ½1 wð12 þ tÞwð2tÞ2
wða2 ðtÞÞ½1 wð12 þ tÞ
wð12 þ tÞ
wðtÞw0 ð2tÞ w0 ðtÞwð2tÞ½1 wð2tÞ
s2 ðtÞ
wðtÞ½1 wð2tÞfwð2tÞ wðtÞ½1 wð2tÞg
wða2 ðtÞÞ½1 wð12 þ tÞ
½GðtÞ s2 ðtÞ;
¼
wð12 þ tÞ
rk e
¼
k¼2
a
e for some
o
1a
1
0oto ;
4
Then, for 0oto14;
w0 12 þ a2 ðtÞ
¼
N
X
1
0ptp ;
4
so w0 ð12 þ tÞ40 because the numerator of w0 ð12 þ tÞ equals
the numerator of G: The condition near 0 ensures that w
is continuous at p ¼ 12; so w is continuous and increasing
on ½0; 34:
For the next extension step to ð34; 78; (2) and 2a2 ðtÞ ¼
1
1
2 þ a1 ðtÞ ¼ 2 þ t give
k ¼ 3; 4; y :
r2 þ r3 þ ? þ rk1 o
255
e40;
so a ð1 aÞðr2 þ r3 þ ? þ rk1 Þ4ð1 aÞe for all k:
It follows that wðð2k 1Þ=2k Þo1 for all k; and wðð2k 1Þ=2k Þ-1 as k-N:
Now
1
3
ð1 aÞ
1
w
w
ow
3ao1 2
4
a
4
1
3w
oa;
4
so wð12Þowð34Þ: For kX3;
1
1
w þ ak ow þ akþ1
2
2
wðakþ1 Þ½1 wð12 þ ak Þ
1
3w þ ak o1 2
wð12 þ ak Þ
1
3wðakþ1 Þow þ ak :
2
so w0 ð12 þ a2 ðtÞÞ40 by (5). Hence, w is continuous and
increasing on ½0; 78: By the proof of Lemma 3, we have
wð12 þ a2 ðtÞÞo1 for 0ptp14: For the following induction,
the same proof gives wð12 þ ak ðtÞÞo1 for 0ptp14 for k ¼
3; 4; y . We omit t to simplify notation.
For kX2; (2) gives
w
wðakþ1 Þ½1 wð12 þ ak Þ
¼
1
;
þ
a
kþ1
2
wð12 þ ak Þ
1
1
0ptp ;
4
w0 ð12 þ akþ1 Þ
¼
wðakþ1 Þw0 ð12 þ ak Þ w0 ðakþ1 Þwð12 þ ak Þ½1 wð12 þ ak Þ
wð12 þ ak Þ2
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
256
1
for 0oto : We claim that for kX1;
4
wðakþ1 Þ½1 wð12 þ ak Þ
w0 12 þ akþ1 ¼
Q
wð12 þ ak Þ kj¼2 wð12 þ aj Þ
"
#
j1
kþ1
Y
X
1
G s2 sj
wð2 þ ai Þ :
j¼3
i¼2
Pkj¼2 wð12
With the understanding that
þ aj Þ ¼ 1
when k ¼ 1; this is true for k ¼ 1 by the preceding paragraph. Suppose it is true for k ¼ 1; y; m:
Then
(6a). Let q ¼ lnð1=pÞ: Then
a
pðpÞ
a
¼ eb½ðqln 2Þ q :
pð2pÞ
As p-0; q-N; and for any constant c and fixed
0oao1 it is known that
lim ½ðq cÞa qa ¼ 0:
q-N
Therefore, limp-0 ½pðpÞ=pð2pÞ ¼ 1: But 1 pð12Þa1; so
(6.1) never satisfies the condition near 0.
w0 ð12 þ amþ2 Þ
¼
¼
wðamþ2 Þw0 ð12 þ amþ1 Þ w0 ðamþ2 Þwð12 þ amþ1 Þ½1 wð12 þ amþ1 Þ
(
wðamþ2 Þ
wð12 þ amþ1 Þ2
wð12 þ amþ1 Þ2
"
#)
j1
mþ1
Y
X
wðamþ1 Þ½1 wð12 þ am Þ
Q
G s2 sj
wð12 þ ai Þ
1
wð12 þ am Þ m
j¼2 wð2 þ aj Þ
i¼2
j¼3
smþ2 wðamþ2 Þ½1 wð12 þ amþ1 Þ
wð12 þ amþ1 Þ
(
)
P Q
wðamþ2 Þ½1 wð12 þ amþ1 Þ wðamþ1 Þ½1 wð12 þ am Þ½G s2 sj Q smþ2
¼
wð12 þ amþ1 Þ
½1 wð12 þ amþ1 Þwð12 þ amþ1 Þwð12 þ am Þ
(
)
Pmþ1 Qj1
wðamþ2 Þ½1 wð12 þ amþ1 Þ G s2 j¼3 sj i¼2 wð12 þ ai Þ
¼
smþ2
Qmþ1 1
wð12 þ amþ1 Þ
j¼2 wð2 þ aj Þ
"
#
j1
mþ2
Y
X
wðamþ2 Þ½1 wð12 þ amþ1 Þ
1
¼ 1
G s2 sj
wð2 þ ai Þ ;
Q
1
wð2 þ amþ1 Þ mþ1
j¼3
i¼2
j¼2 wð2 þ aj Þ
so it is true also for k ¼ m þ 1: At this point in the
induction process, we know that w is continuous
and increasing on ½0; 12 þ amþ1 ð14Þ ¼ ½0; 12 þ amþ2 ð0Þ
with 0owð12 þ aj Þo1 for jpm þ 1: Thus, by (5) and
0owð12 þ aj Þo1 for jpm þ 1; we have
G4
N
X
k¼2
sk 4s2 þ
mþ2
X
j¼3
sj
j1
Y
wð12 þ aj Þ;
i¼2
so w0 ð12 þ amþ2 ðtÞÞ40 for 0oto14:
It follows by induction that w extended is continuous
and increasing on ½0; 1Þ: In addition, wðpÞ-1 as p-1 by
the proof of Lemma 3. &
Proof of Theorem 4. As noted earlier, we assume that p
is an increasing function from ½0; 1 onto ½0; 1: This is
true for each of (6a) through (6f). We focus on the
condition near 0, i.e.,
pðpÞ
¼ 1 pð12Þ;
lim
p-0 pð2pÞ
and show that it fails except for noted special cases.
(6b). We have
1=g
pðpÞ
1 ð2pÞg þ ð1 2pÞg
1
¼ g
- g
g
g
pð2pÞ 2
2
p þ ð1 pÞ
as p-0:
Moreover,
pð12Þ ¼ 1=ð2gþ1=g1 Þ;
and
lim½pðpÞ=pð2pÞ ¼ 1 pð12Þ if and only if
therefore
ð21=g1 Þð2g 1Þ ¼ 1:
This holds for g40 if and only if g ¼ 1; and when g ¼ 1;
p is the identity.
(6c). As in the preceding case, lim½pðpÞ=pð2pÞ ¼ 1=2g :
We also have pð12Þ ¼ a=ða þ 1Þ; so the condition near 0
holds if and only if a ¼ 2g 1: Because
1 pð1 pÞ
1 að1 2pÞg þ ð2pÞg
1
¼ lim g
lim
¼ g;
p-0 1 pð1 2pÞ
p-0 2
2
að1 pÞg þ pg
the condition near 1 requires 1=2g ¼ a=ða þ 1Þ; i.e., a ¼
1=ð2g 1Þ: It follows that the conditions near 0 and near
1 hold if and only if a ¼ g ¼ 1; in which case p is the
identity.
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
It follows that w is differentiable at
holds. &
(6d). Here
a
pðpÞ
a
¼ eðb=aÞ½p ð2pÞ ; aa0; b40:
pð2pÞ
If a40 then lim½pðpÞ=pð2pÞ ¼ 1: If ao0 then
lim½pðpÞ=pð2pÞ ¼ 0: Because 0opð12Þo1; the condition
near 0 never holds.
(6e). We have pðpÞ=pð2pÞ ¼ 1=2b and pð12Þ ¼ 1=2b ; so
1=2b ¼ 1 1=2b 32b1 ¼ 13b ¼ 1; and p is the identity when b ¼ 1:
(6f). Let q ¼ lnð1=pÞ: Then
pðpÞ
ð1 a ln pÞb
¼
¼
pð2pÞ ð1 a ln 2pÞb
1 þ aq a ln 2
1 þ aq
257
1
2
if and only if (7)
Proof of Lemma 5. Because the numerator and denominator of (A.1) both approach 0 as t-0; we can use
l’Hospital’s rule on (A.1) to conclude that
w0 ð12 þ tÞ ¼ fw000 ðtÞwð2tÞ½wð2tÞ 1 þ w00 ðtÞ½2wð2tÞw0 ð2tÞ
w0 ð2tÞ þ 2w00 ðtÞwð2tÞw0 ð2tÞ þ 2w0 ðtÞw0 ð2tÞ2
þ 2w0 ðtÞwð2tÞw00 ð2tÞ þ w0 ðtÞw00 ð2tÞ
þ wðtÞw000 ð2tÞg=f2w0 ð2tÞ2 þ 2wð2tÞw00 ð2tÞg
for t-0: When t-0; this reduces to
1 w00 ð0Þ
þ
:
w0 ð12 Þ ¼ w0 ð0Þ þ
4 w0 ð0Þ
b
:
As p-0; q-N; so limp-0 ½pðpÞ=pð2pÞ ¼ 1: Because
pð12Þ40; the condition near 0 never holds. &
This and the final equation in the proof of Lemma 4
yield w00 ð0Þ ¼ ½3 4wð12Þw00 ð0Þ; which is tantamount to
the conclusion of Lemma 5. &
Proof of Lemma 4. Assume Axiom 2. By definition, w is
þ
differentiable at 12 if and only if w0 ð12 Þ ¼ w0 ð12 Þ: The
0 1
value of w ð2 Þ is obtained from the specification of w on
þ
½0; 12 as in Axiom 1. Our evaluation of w0 ð12 Þ is based on
(2) for w in the form
Proof of Lemma 6 (Outline). When the first expression
for w0 ð12 þ tÞ in the proof of Lemma 4 is differentiated in
the usual way, we get
wðtÞ½wð2tÞ 1
w 12 þ t ¼ 1 þ
wð2tÞ
þ 2w0 ðtÞw0 ð2tÞwð2tÞ 2wðtÞw0 ð2tÞ2 g=wð2tÞ3 :
for 0oto12:
According to our convention for derivatives following
the statement of Axiom 1, the limits of w0 ð2tÞ and w00 ð2tÞ
as t-0 are 2w0 ð0Þ and 4w00 ð0Þ; respectively.
The preceding equation yields
w0
1
w0 ðtÞwð2tÞ½wð2tÞ 1 þ wðtÞw0 ð2tÞ
þt ¼
:
2
wð2tÞ2
Because the numerator and denominator both approach
0 as t-0; we use l’Hospital’s rule to get
w0 ð12
w00 ðtÞwð2tÞ½wð2tÞ 1 þ 2w0 ðtÞwð2tÞw0 ð2tÞ þ wðtÞw00 ð2tÞ
þ tÞ ¼
2wð2tÞw0 ð2tÞ
ðA:1Þ
w00 ðtÞ½wð2tÞ 1 w00 ð2tÞ wðtÞ
¼ w ðtÞ þ
þ
2w0 ð2tÞ
2w0 ð2tÞ wð2tÞ
0
w00 ð12 þ tÞ ¼ fw00 ðtÞ½wð2tÞ3 wð2tÞ2 þ w00 ð2tÞwðtÞwð2tÞ
ðA:2Þ
As t-0; the numerator and denominator both go to 0,
so we use l’Hospital’s rule and find that the differentiated numerator and differentiated denominator both
go to 0 as t-0: We therefore use l’Hospital’s rule again.
The ratio of the twice-differentiated numerator to the
twice-differentiated denominator of the preceding form
of w00 ð12 þ tÞ is then evaluated as t-0 with the aid of the
þ
condition near 0 and wðjÞ ð2tÞ-2j wðjÞ ð0Þ to obtain w00 ð12 Þ
equal to the right side of the displayed equation of
Lemma 6. Because w is second differentiable at 12 if
þ
and only if w00 ð12 Þ ¼ w00 ð12 Þ; the lemma’s conclusion
follows. &
Proof of Theorem 5. Let bk ¼ ð2k 1Þ=2k for k ¼
1; 2; y . By (1) for w with ð1 þ pÞ=2 ¼ bkþ1 þ t; we have
wðbkþ1 þ tÞ ¼ 1 wð12bk þ tÞ½1 wðbk þ 2tÞ
wðbk þ 2tÞ
for all t in a small open interval that contains 0.
Differentiation gives
w0 ð12bk þ 1Þ
wðbk þ 2tÞ
0
w ðbk þ 2tÞwð12bk þ tÞ
þ
:
wðbk þ 2tÞ2
w0 ðbkþ1 þ tÞ ¼ w0 ð12bk þ tÞ for t-0: The condition near 0 says
wðtÞ=wð2tÞ-1 wð12Þ as t-0; so (A.2) yields
þ
w0 ð12 Þ
00
3
1 w ð0Þ
¼ w ð0Þ þ wð2Þ 0 :
4
w ð0Þ
0
that
Suppose w is differentiable at bj for jpk: Then w is
differentiable on ½0; bkþ1 Þ; so the right side of the
258
D.E. Bell, P.C. Fishburn / Journal of Mathematical Psychology 47 (2003) 244–258
preceding expression is continuous in t for t in a small
open interval that contains 0, and it follows that w is
differentiable at bkþ1 : Since we assume that w is
differentiable at b1 ¼ 12; induction on k implies that w
is differentiable at all bk : &
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