Making Rank-Dependent Utility
Tractable for the Study of
Ambiguity
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comments invisible.
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Peter P. Wakker, June 16, 2005
MSE, Université de Paris I
Aim:
Make rank-dependent utility
tractable to a general public and
specialists alike, in particular for
ambiguity.
Tool:
Ranks!
Spinoff: Some changes of minds:
2
Question 1 to audience:
From what can we best infer that people
deviate from EU for risk (given
probabilities)?
a. Allais paradox.
b. Ellsberg paradox.
c. Nash equilibria.
3
Question 2 to audience:
From what can we best infer that people
deviate from SEU for uncertainty (unknown
probabilities)?
a. Allais paradox.
b. Ellsberg paradox.
c. Nash equilibria.
4
Question 3 to audience:
Assume rank-dependent utility for unknown
probabilities (Choquet Expected utility).
From what can we best infer that
nonadditive measures are convex (=
superadditive)?
a. Allais paradox.
b. Ellsberg paradox.
c. Nash equilibria.
After this lecture:
Answer to Question 1 ("nonEU for risk") is:
Allais paradox.
Answer to Question 3 ("capacities convex in
RDU = CEU") is:
Allais paradox!
Answer to Question 2 ("nonEU for
uncertainty") is:
both Allais and Ellsberg paradox.
P.s.: I do think that the Ellsberg paradox has
more content than the Allais paradox.
Explained later.
5
Other change of mind:
6
The inequality
Decision under risk  Decision under uncertainty
in the strict sense of
[ Decision under risk

Decision under uncertainty
=]
is incorrect!
Decision under risk

Decision under uncertainty
!
That's how it is!
7
Outline of lecture:
1. Expected Utility for Risk.
2. Expected Utility for Uncertainty.
3. Rank-Dependent Utility for Risk, Defined through
Ranks.
4. Where Rank-Dependent Utility Differs from
Expected Utility for Risk.
5. Where Rank-Dependent Utility Agrees with
Expected Utility for Risk, and some properties.
6. Rank-Dependent Utility for Uncertainty, Defined
through Ranks.
7. Where Rank-Dependent Utility Differs from
Expected Utility for Uncertainty as it Did for Risk.
8. Where Rank-Dependent Utility Agrees with
Expected Utility for Uncertainty.
9. Where Rank-Dependent Utility Differs from Expected Utility for Uncertainty Differently than for Risk.
10. Applications of Ranks.
At 1 and 2: They
know well. Lines
and notation may
give new insights
Line of DUR 
DUU, and of first
seeing how to
measure the
subjective
concepts in a
theory, and then
how to axiomatize
the theory. For
instance, this
can't be done yet
for multiple priors.
8
1. Expected Utility for Risk
(p1:x1,…,pn:xn) =
p1
.
.
.
pn
x1
..
.
xn
is prospect yielding €xj with probability pj, j=1,…,n.
Expected
utility:
p1
.
.
.
pn
x1
..
.  p1U(x1 ) + ... + pnU(xn)
xn
9
U: subjective index of risk attitude
(watch out: only under expected utility!!!!!)
How measure U from preferences?
Set U() = 1, U() = 0.
Find, for each     , probability p such that
p

~
1–p
Then U() = p.


Assume following data deviating from expected value
0.10
€1
~
0
0.50 €100
€100
€ 9~
0.90
(a)
0.30
€100
€25~
0.70
(b)
0
(c)
0.70
0
(d)
€100
€81~
€49 ~
0.50
0.90
€100
10
0.30
0
(e)
0.10
0
U(100) = 1, U(0) = 0
EU:
U(1)
=
0.10U(100)
EU:
EU: U(x)
U(9)==pU(100) = p.
Psychology:1x==w(.10)100
w(p)100
Psychology:
+
0.90U(0)
=
0.10.
Here
is graph
U(x):
0.30U(100)
= of
0.30.
Here is graph of w(p):
p 1
€
€100
(e)
(e)
0.7
€70
(d)
(d)
(c)
(a)
0
€0
next p.
€30
(c)
€30
(b)
0.3
€70
€100
€
€0
0
(a) (b)
0.3
1
0.7
p
go to
p.27,RDU
p
Format  ~
1–p
11


has empirical problems:
Certainty effect!
Alternative format (McCord & de Neufville '86)
q
1–q

Q
q
p


1–p
~
1–q
Q
Consistency in utility measurement (substition):
Upper and lower p should be the same.
= substitution for 1&2-outcome prospects.
 vNM independence.
12
Theorem.
Expected Utility

(a) Continuity in probabilities;
(b) monotonicity;
(c) weak ordering;
(d) consistency in utility measurement
("substitution").

13
Proof.
p1
p1
x1
.
.
.
pj
xj
~
pj
.
.
.
pn
x
n
x1
.
.
.
pn
.
.
.
xn
U(x1)
p1
U(xj )

1U(xj ) 
~
pj
.
.
.
.
.
.
pn

1U(x1) 
U(xj )

1U(xj ) 
U(xn)

1U(xn)

p1U(x1)+ . . . +pnU(xn)

=
1 p1U(x1) . . .  pnU(xn)

14
Well-known implication: Independence
from common consequence ("sure-thing
principle"):
qr
qr


1–q
qr
P
1–q


qr

1–q
next p.
P
1–q

Q

Q
go to p. 34,
where RDU =
EU for risk
15
Well-known violation: Allais paradox.
M: million €
w
.89
0
.10
.01
0
.10 b
w

5M
1M
.10
.89
1M
w
.01
0
b
.10
.10

5M
Is the certainty effect.
next p.
.89
0
.10
.01
1M
.10 b
.89
w 1M
.01
1M
b
.10
>
< EU
1M
OK for RDU.
go to p. 33,
where RDU 
EU for risk
16
In preparation for rank-dependence and
decision under uncertainty, remember:
In (p1:x1,…,pn:xn), we have liberty to
choose x1  ...  xn.
2. Expected Utility for Uncertainty
Wrong start for DUU:
Let S = {s1,…,sn} denote a finite state space.
x : S   is an act, also denoted as an ntuple x = (x1,…,xn).
Is didactical mistake for rank-dependence!
Why wrong?
Later, for rank-dependence, ranking of
outcomes will be crucial. Should use
numbering of xj for that purpose; as under
risk! Should not have committed to a
numbering of outcomes for other reasons.
So, start again:
17
18
S: state space, or universal event.
Act is function x : S   with finite range.
x = (E1:x1, …, En:xn): yields xj for all sEj, with:
x1,…,xn   are outcomes.
E1, …, En are events partitioning S.
No commitment to a numbering of outcomes!
As for risk.
Important notational point for rank-dependence
(which will come later).
If E1,…,En understood, we may write (x1, …,xn).
(E1:x1,…,En:xn) =
Subjective
Expected
utility:
E1
.
.
.
En
E1
.
.
.
En
19
x1
..
.
xn
x1
..
.  (E1)U(x1 ) + ... + (E1)U(xn)
xn
U: subjective index of utility.
: subjective probability.
How measure these? Difficult, because two
unknown scales.
If can measure one, then other is easy (Ramsey).
20
First measure :
Savage (1954), Abdellaoui & Wakker (2005).
First measure U:
Several papers. Is our approach today.
21
Notation:  Rx is (x with outcomes on E
E
replaced by ):and ranking position of E is R
10E1x = (E1:10,E2:x2,.., En:xn);
Enx = (E1:x1,.., En-1:xn-1, En:);
etc.
Monotonicity:     Ex  Ex;
next p.
next p.
R
22
If there exist x,y, nonnull E with:
ERx ~ ERy
and
then
 ERx ~ ERy
 ~*r 
rank-dependent
Lemma. Under (subj) expected utility,
 ~r*   U() – U() = U() – U() .
This is how we measure U under SEU.
Need not know !
next p.
next p.
23
If
 ~*r  and ' ~*r  for ' > ,
then, under SEU, RDU
U() – U() = U() – U() and
U(') – U() = U() – U():
Inconsistency!
rankTradeoff consistency precludes such
inconsistencies. That is:
improving any outcome in a ~*r relationship
breaks the relationship.
next p.
next p.
Theorem.
The following two statements are equivalent:
24
rank-dependent
(i) (cont. subj) expected utility.
(ii) four conditions:
(a) weak ordering;
(b) monotonicity;
(c) continuity;
(d) tradeoff consistency.
rank-
Tradeoff consistency also gives !
Inconsistencies in those generate such in ~*.
r
next p.
next p.
25
Well-known implication: sure-thing principle:
ERx  ER y
rank-

 ERx   ER y
next p.
go to p.42,
RDU=EU
26
Almost-unknown
implication
(Not-so-well-known) violation (MacCrimmon &
Larsson '79; here Tversky & Kahneman '92).
Within-subjects expt, 156 money managers.
d: DJtomorrw–DJtoday. L: d<30 ; M: 30d35; H: d>35;
K: $1000.
w
w
L
L
MH
Hb
0
0 
75K
H
L
w
M
b
H
25K
0 
75K
MH
Hb
LH
w
M
b
H
0
25K (77%)
25K
>
< SEU
25K
25K (66%)
OK for RDU:
25K pessimism.
Certainty-effect & Allais hold for uncertainty in
general, not only for risk!
go to p.41,
RDUEU
27
3. Rank-Dependent Utility for Risk,
Defined through Ranks
Empirical findings: nonlinear treatment of
probabilities. Hence RDU. Two steps for
to public that the two steps go together,
getting the theory. Explain
and in isolation are vacuous. Only jointly they
constitute a decision theory.
Step 1. Deviations from expected value in
Section 1: nonlinear perception/processing of
probability, through w(p).
go to p. 10,
with ut.curv.
Step 2. Turn this into decision theory through
rank-dependence.
Rank-dependent utility of
p1
.
.
.
pn
28
x1
..
. ?
xn
First rank-order x1 > … > xn.
Decision weight of xj will depend on:
1. pj;
2. pj–1 + … + p1, the probability of receiving
something better. The latter will be called
a rank.
So, rank-dependent utility of
p1
.
.
.
pn
29
x1
..
. ?
xn
First rank-order x1 > … > xn.
Then rank-dependent utility is
1U(x1 ) + … + nU(xn)
where
j = w(pj + pj–1 + … + p1) – w(pj–1 + … + p1).
The decision weight j
depends on pj and on pj–1 + … + p1:
pj–1 + … + p1 is the rank of pj,xj, i.e. the
probability of receiving something better.
30
Ranks and ranked probabilities (formalized
hereafter) are proposed as central concepts
in this lecture.
Were introduced by Abdellaoui & Wakker
(Theory and Decision, forthcoming in July
2005).
With them, rank-dependent life will be much
easier than it was ever before!
31
In general, pairs pr, also denoted p\r, with p+r 
1 are called ranked probabilities.
r is the rank of p.
(pr) = w(p+r) – w(r) is the decision weight of pr.
Again, rank-dependent utility of
with rank-ordering x1  …  xn:
p1
.
.
.
pn
32
x1
..
.
xn
(p1r1)U(x1) + … + (pnrn)U(xn)
with rj = pj–1 + … + p1 (so r1 = 0).
The smaller the rank r in pr, the better the
outcome.
The best rank, 0, is also denoted b, as in pb = p0.
The worst rank for p, 1–p, is also denoted w, as
in pw = p1–p.
33
4. Where Rank-Dependent Utility Differs
from Expected Utility for Risk
go to p. 15,
with Allais
Allais paradox explained by rank dependence.
Now the expression rank dependence can be
taken literally!
34
5. Where Rank-Dependent Utility Agrees
with Expected Utility for Risk, and
Properties
go to p. 14,
with risks.th.pr
Common consequence implication of EU
goes through completely for RDU if we
replace probability by ranked probability.
Some properties, suggested by Allais
paradox, follow now (more to come later).
Now see Fig. of w-shaped.doc
35
w convex (pessimism):
r < r'  w(p+r) – w(r)  w(p+r') – w(r')
Equivalent to:
(pr) increasing in r.
Remember: big rank is bad outcome.
"Decision weight is increasing in rank."
w concave (optimism) is similar.
2 more pessimistic than 1, i.e. w2 more
convex than w1:
r < r', 1(pr) = 1(qr')  2(pr)  2(qr')
Inverse-S
w+
36
1
0 g
0
b1 p
good-outcome region insensitivity- bad-outcome region
region
Good probs weighted morer than in insensitive region:
(pb)  (pr) on [0,g]  [g,b]
Bad probs weighted more than in insensitive region:
(pw)  (pr) on [g,b]  [b,1].
6. Rank-Dependent Utility for
Uncertainty, Defined through Ranks
x = (E1:x1, …, En:xn) was act, with Ej's
partitioning S.
Rank-dependent utility for uncertainty: (also
called Choquet expected utility)
W is capacity, i.e.
(i) W() = 0;
(ii) W(S) = 1 for the universal event S;
(iii) If A  B then W(A)  W(B) (monotonicity
with respect to set inclusion).
37
38
ER, with ER = , is ranked event, with R
the rank.
(ER) = W(ER) – W(R) is decision weight
of ranked event.
RDU of x = (E1:x1, …, En:xn),
with rank-ordering x1  …  xn,
is:
jn (EjRj)U(xj)
with Rj = Ej–1  …  E1 (so R1 = ).
Compared to SEU, ranks Rj have now been
added, expressing rank-dependence.
39
The smaller the rank R in ER, the better the
outcome.
The best (smallest) rank, , is also denoted
b, as in Eb = E.
The worst (biggest) rank for E, Ec, is also
c
w
E
denoted w, as in E = E .
40
Difficult notation in the past:
S = {s1,…,sn}. For RDU(x1,…,xn),
take a rank-ordering r of s1,...,sn such that
xr1  ...  xrn.
For each state srj,
rj = W(srj,…, sr1) – W(srj-1,…, sr1)
RDU = r1U(xr1) + … + rnU(xrn)
Due to r-notation, difficult to handle.
r(2) = 5: Is state s2 fifth-best, or is state s5
second-best? I can never remember!
41
7. Where Rank-Dependent Utility Differs
from Expected Utility for Uncertainty as
It Did for Risk
Convexity of W follows from Allais paradox!
Easily expressable in terms of ranks!
Comment on double
exclamation marks.
go to p. 26,
Allais for
uncertainty
42
8. Where Rank-Dependent Utility Agrees
with Expected Utility for Uncertainty
The whole measurement of utility, and
preference characterization, of RDU for
uncertainty is just the same as SEU, if we
simply use ranked events instead of events!
go to p. 21, TO
etc.
9. Where Rank-Dependent Utility Differs
from Expected Utility for Uncertainty
Differently than It Did for Risk
Allais: deviations from EU.
Pessimism/convexity of w/W,
or insensitivity/inverse-S.
For risk and uncertainty alike.
Deviations from EU in an absolute sense.
Ellsberg: more deviations from EU for
uncertainty than for risk.
More pessimism/etc. for uncertainty than for
risk.
Deviations from EU in an relative sense.
Deviations from EU: byproduct.
43
44
Historical coincidence:
Schmeidler (1989) assumed EU for risk, i.e.
linear w.
Then:
more pessimism/convexity for uncertainty than
for risk (based on Ellsberg),

pessimism/convexity for uncertainty.
Voilà source of numerous misunderstandings.
45
Big idea to infer of Ellsberg is not, I think,
ambiguity aversion.
Big idea to infer from Ellsberg is, I think,
within-person between-source comparisons.
Many, even prominent, economists haven't yet
caught up on this point.
Not possible for risk, because risk is only
one source. Typical of uncertainty, where
there are many sources.
Uncertainty is rich domain, with no patterns
to be expected to hold in great generality.
In this rich domain, many phenomena are
present and are yet to be discovered.
10. Applications of Ranks
General technique for revealing orderings
(AR)  (BR') from preferences: Abdellaoui
& Wakker (2005, Theory and Decision).
Thus, preference foundations can be given
for everything written hereafter.
46
What is null event?
Important for updating, equilibria, etc.
(Eb) = 0.
E is null if W(E) = 0?
E is null if W(Ec) = 1? (Ew) = 0.
47
E is null if (H:, E:, L:) ~ (H:, E:, L:)?(EH) = 0.
For some   , H …?
Or for all   , H …?
?
We: Wrong question! Better refer to ranked
events!
Plausible condition is null-invariance:
independence of nullness from rank.
W convex:  increases in rank.
W concave:  decreases in rank.
W symmetric: (Eb) = (Ew).
Inverse-S: There are
[G,B], event-region of insensitivity; with
[,G] good-event region,
[B,S] the bad-event region.
Good-event inequality (weighing good events
better than insensitive):
(Eb)  (ER) on event-interval [,G]  [G,B]
and bad-event inequality (weighing bad events
better than insensitive):
(Ew)  (ER) on event-interval [G,B]  [B,S].
48
49
Theorem.
2 is more ambiguity averse than 1 in sense
that W2 is more convex than W1
iff
1(BR') = 1(AR)
with R'  R

2(BR')  2(AR).
50
Theorem.
Probabilistic sophistication holds

[(AR)  (BR)  (AR')  (BR')].
In words: ordering of likelihoods is
independent of rank.
Updating on A given B, with A  B.
What is W(A) if B is observed?
b)
(A
W(A) .
Gilboa (1989a,b):
b)
(B
W(B)
51
c
B
(A )
–
.
Dempster & Shafer:
1 – W(Bc)
(Bw)
W(A)
.
Jaffray, Denneberg:
W(A) + 1 – W((B\A)c)
Gilboa & Schmeidler (1993):
(Ab)
depends on optimism /
b) + ((B\A)w)
(A
pessimism. Ranks
Cohen,Gilboa,Jaffray,&Schmeidler
formalize this.
W(ABc)
W(Bc)
(2000): lowest one did best.
52
Conclusion:
With ranks and ranked probabilities or
events, rank-dependent utility becomes
considerably more tractable.
53
Le fin!