Updating Ambiguity Averse
Preferences
Eran Hanany
Tel Aviv University
Faculty of Engineering
Peter Klibanoff
Northwestern University
Kellogg School of Management
Managerial Economics and
Decision Sciences (MEDS)
Introduction
• Concerned with how one should update preferences upon
receiving new information.
Approach will be to investigate update rules delivering the
following properties:
• Dynamic Consistency
– Ex-ante optimal information contingent choices are respected by
updated preferences
• Closure
– If limit consideration ex-ante to a certain model, then updated
preferences should remain within that model
• Update beliefs, not tastes (when separated)
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Introduction
• In Hanany and Klibanoff (2007), we pursued these issues
in the context of a specific model of preferences:
Maxmin expected utility (Gilboa and Schmeidler 1989)
• In the current paper, we provide a characterization of
dynamically consistent update rules that applies to a
significantly broader set of preferences.
• We also apply this general result to some specific models
of recent interest in the ambiguity literature, while
delivering the first rules for updating these models in a
consistent way.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Closely Related Literature
• Non-expected utility updating and consistency (Machina
1989, McClennen 1990, Machina and Schmeidler 1992,
Epstein and Le Breton 1993, Eichberger and Grant 1997,
Segal 1997, Wakker 1997, Grant, Kajii, Polak 2000)
• DC and updating MEU preferences (Hanany and
Klibanoff 2007)
• Recursive Smooth Ambiguity preferences (Klibanoff,
Marinacci, Mukerji 2006)
• Dynamic Variational preferences (Maccheroni, Marinacci,
Rustichini 2006) (includes Recursive MEU (Epstein and
Schneider 2003
• Consistent Planning under ambiguity (Siniscalchi 2006)
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Preferences
• Anscombe-Aumann framework
– Objects of choice are acts,
f :S  L
• Consider non-degenerate, continuous, monotonic
preference relations that, when restricted to constant acts,
obey the vNM expected utility axioms.
V (u  f )
• V is quasi-concave if and only if preferences satisfy
Schmeidler's Uncertainty (Ambiguity) Aversion axiom:
f h

f  (1   )h  h
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Preferences (cont.)
• Early models satisfying ambiguity aversion include the
now-classic:
– Maxmin Expected Utility (Gilboa and Schmeidler 1989)
– concave Choquet Expected Utility (Schmeidler 1989); concave
Cumulative Prospect Theory over gains (Tversky and Kahneman,
1992)
• Recent advances have led to several more flexible
models of ambiguity aversion, including:
– concave Smooth Ambiguity preferences (Klibanoff, Marinacci,
Mukerji 2005; Nau 2006; Seo 2006)
– Variational preferences (Maccheroni, Marinacci, Rustichini 2006)
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
• Quadruple:
Update Rules
– A preference representation, (V , u )
– A convex, compact feasible set of acts, B
– An (interior) act, g, optimal in B
– A non-null event, E  S
• An update rule is a function defined on a set of such
quadruples that maps each quadruple to a new
preference representation (VE , g , B , u E , g , B ) that makes Ec a
null event.
• Restrict attention to rules for which updated preferences
are ambiguity averse (VE , g , B quasiconcave) and risk
preferences are unchanged u E , g , B  u .
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Dynamic Consistency of an Update Rule
“An ex-ante optimal act cannot be feasibly improved
conditional on any planned-for event.”
Formally (Hanany & Klibanoff 2007):
DC: For each quadruple in the domain of the update rule,
VE , g , B (u  g )  VE , g , B (u  f ) for all f  B with f  g on E c .
• Preferences for acts other than g need not be preserved
• Comparison with feasible acts identical on E-complement
• Helps separate dynamic consistency from consequentialism
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
A General Characterization
Definition: For a quasiconcave V, the measures supporting the
conditional indifference curve at h on E are:
TE,h(V) ≡ {q∈Δ(E) ∫(u∘f)dq ≥ ∫(u∘h)dq for all f such that V(u∘f)>V(u∘h)}.
Definition: The measures supporting the conditional optimality of
h are:
QE,h,B ≡
{q∈Δ∣q(E)>0 and ∫(u∘h)dq ≥ ∫(u∘f)dq for all f∈B with f=h on Ec}.
Denote by QE,h,B(E) the set of Bayesian conditionals on E of measures
in QE,h,B.
Theorem: An update rule satisfies DC if and only if
TE,g(VE,g,B) ∩ QE,g,B(E) ≠ 
for each quadruple in the domain of the update rule.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
A General Characterization
•
This gives us a test for dynamic consistency of
an update rule for any quasiconcave model:
1. Calculate TE,g(VE,g,B)
2. Calculate QE,g,B(E)
3. See if they intersect.
•
•
If VE,g,B is concave, TE,g(VE,g,B) can be replaced
by a set of measures derived from the
superdifferential of VE,g,B at u  g . If
differentiable then look at gradient.
Similarly, smoothness of the relevant slice of B
at u  g simplifies calculation of QE,g,B(E).
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Application: Updating
Smooth Ambiguity preferences
Definition: A smooth ambiguity preference (KMM 2005)
over acts has the following representation:
V (u  f )  
 


  (u  f )d d
where  is an increasing real-valued function and captures
ambiguity attitude ( concave = ambiguity aversion) and 
is a subjective probability over probability measures  on
the state space. Ambiguity is captured through
disagreement in the support of  about the probability of
an event.
.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
DC updating of SA preferences
•
Assume  differentiable. Consider update rules
that:
1. Leave  unchanged
2. Do not depend on the feasible set B given (,u,,E,g).
Theorem: Such rules are dynamically consistent iff
 E  E ,g [ ' (E E u  g ) E ( s)] 


 E  [ ' (E u  g )] 
E ,g
E


sE
 E  [ ' (E u  g ) ( s)] 

 
 E [ ' (E u  g ) E ] 

 
 sE
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Application: Expected Utility
Let’s take a step back – to expected utility (EU).
Expected utility corresponds to the special case
where  is affine, so ’ is constant. The only beliefs
that matter in this case are the reduced measures
p = E  and pE,g = E  E
Corollary
U is dynamically consistent iff pE,g is the Bayesian
update of p.
E ,g
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Smooth Ambiguity
• Consider again ambiguity averse Smooth
Ambiguity (SA) preferences.
• Will Bayesian updating work here?
• No! Easy to show in a simple dynamic Ellsberg
example. Also follows from our Theorem.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Smooth Ambiguity
• Since Bayesian updating is dynamically
inconsistent, to find a rule that satisfies
consistency for SA preferences like Bayes’ rule
does for EU preferences we need to look
elsewhere.
• We will look at reweightings of Bayes’ rule.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Reweighted Bayes’ Rules
Beliefs conditiona l on event E :
*

̂ ( , s)   ( ) ( s) ( , , u, g , E )
  , s  E,
 ( ) ( E )  ( ,  , u, g , E )
 E, g( )  ̂ ( , E ) 
E  ˆ ( E ) (ˆ ,  , u, g , E )
*
E, g
̂
 E*, g ( , s )  ( s )
 E (s)  *

̂ E, g( , E )  ( E )
Conditiona l preference :
f  E, gh

E  E, g  (E E u  f )  E  E, g (E E u  h)
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Smooth Ambiguity
Proposition
A reweighted Bayes’ rule is dynamically
consistent iff E,g is the measure generated by
setting
' (E u  g )
 ( ,  , u, g , E ) 
' (E E u  g )
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Properties of the Smooth Rule
• This rule generalizes Bayesian updating of
beliefs.
• Exactly coincides with Bayesian updating when:
– preferences are ambiguity neutral, or
– the ex-ante optimum, g, is constant in utilities.
In both cases the distortion factor vanishes.
• Departs from Bayesian updating in a manner that
depends on:
– The ex-ante optimum, g.
– the ambiguity attitude of the decision-maker
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Properties of the Smooth Rule
• Compared to Bayes’ Rule, our rule overweights
measures that result in higher conditional
valuations of g relative to unconditional
valuations of g.
 ' (E u  g )
 E , g ( )   ( ) ( s)
 ' (E u  g )
E
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Properties of the Smooth Rule
The Smooth rule is commutative. Commutativity
means that the order of information received
does not affect updating.
Specifically, for any non-null events E and F
having non-null intersection:
( E , g ) F , g   E F , g  ( F , g ) E , g
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Conclusion
• General result characterizing dynamic consistency for
convex (ambiguity/uncertainty averse) preferences.
• Provide characterization of dynamically consistent
updating for ambiguity averse Smooth Ambiguity
preferences and identify a natural generalization of Bayes’
rule.
• Provide characterization of dynamically consistent
updating for Variational preferences (also have further
results on specific rules). Show Bayesian updating is
consistent for Multiplier preferences.
• Substantially generalizes previous results of ours on
updating MEU. E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
The End
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Properties of the Smooth Rule
The Smooth rule also satisfies a strict version of
dynamic consistency when there is strict
ambiguity aversion (  is strictly concave):
• Strict DC: Add to DC that if g is ex-ante strictly
preferred (indifferent) to some f  B
with f  g on E c, then g remains strictly preferred
(indifferent) to f after learning E.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Smooth Ambiguity Model (cont.)
Only model of decision-making under ambiguity
currently available that simultaneously allows:
1. separation of ambiguity attitude from perception
of ambiguity
2. flexibility in ambiguity attitude
3. subjective and flexible perception of which
events are ambiguous
4. the tractability of smooth preferences
5. expected utility as a special case for any given
ambiguity attitude.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
3-Color Ellsberg Paradox
30 +
Black
?
Red
+
?
=
Yellow
Bet on Black
1
0
0
Bet on Red
0
1
0
Bet against Black
0
1
1
Bet against Red
1
0
1
Bet on Black > Bet on Red
Bet against Black > Bet against Red
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
90
Dynamic Consistency vs.
Recursion
Assume (1,0,0) ≻ (0,1,0) and (0,1,1) ≻ (1,0,1) as in Ellsberg
b
E
B
(1,0,·)
Ec
r
E
B
(·,·,0)
R
(0,1,·)
b'
Ec
r'
R
(·,·,0)
(1,0,0)
(0,1,0)
Dynamic Consistency vs.
Recursion
Assume (1,0,0) ≻ (0,1,0) and (0,1,1) ≻ (1,0,1) as in Ellsberg
b
E
B
(1,0,·)
Ec
r
E
B
(·,·,1)
R
(0,1,·)
b'
Ec
r'
R
(·,·,1)
(1,0,1)
(0,1,1)
Dynamic consistency vs.
inconsistency
Assume (1,0,0)≻(1-ε,-ε,-ε)≻(ε,1+ε,ε)≻(0,1,0) in Ellsberg
d
r+
b-
B
(1,0,·)
E
Ec
E
R
B or R
(0,1,·) (·,·,0)
B
(1-,-,·)
Ec
E
B or R
(·,·,-)
R
(,1+,·)
Ec
B or R
(·,·,)
Variational preferences
Definition: A variational preference (MMR 2006)
over acts has the following representation:
V (u  f )  min
p
 (u  f )dp  c( p)
where u is a vN-M utility function and c:Δ→[0,∞] is
grounded, convex and lower semicontinuous.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Variational preferences
Special cases include:
• Maxmin EU (Gilboa and Schmeidler 1989)
 0 if p  C
c( p)  
 V (u  f )  min
 if p  C
•
pC
 (u  f )dp
Robust Control Preferences (Hansen and
Sargent 2001)

 dp 
 if p  q
E p  ln
c( p)  
 dq 

 elsewhere

E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
DC updating of Variational
preferences
•
Consider update rules that:
1. Leave u unchanged, and
2. Make Ec a null event.
Proposition: Such rules are dynamically consistent
iff
arg min
p
 (u  g)dp  c

E, g ,B
(
p
)

Q
( E)  
E, g ,B
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Expected Utility
Q: Why use Bayes’ rule to update beliefs under
EU?
A: It is the unique such update rule under EU that is
dynamically consistent.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Bayesian Update Rule
Beliefs conditiona l on event E :
  , s  E,
 * ( , s )   ( ) ( s )
 E ( )   E* ( , E ) 
 ( ) ( E )
E  ˆ ( E )
 E* ( , s )  ( s )
 E (s)  *

 E ( , E )  ( E )
Conditiona l preference :
f E h

E  E  (E E u  f )  E  E  ( E E u  h )
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Smooth Ambiguity
• Unfortunately, Bayesian updating is not
dynamically consistent for SA preferences.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Proof Sketch
Part 1: Smooth Rule satisfies DC
• Under concavity & differentiability of  and
convexity of the feasible set, optimality of g is
equivalent to a unique hyperplane separating the
upper contour set from the feasible set at g.
• DC requires conditional optimality of g
qg
among feasible acts agreeing with g on
g
Ec.
• Sufficient to show the gradients of the
unconditional and conditional representations are
proportional on E at g.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Smooth Rule satisfies DC (cont.)
For s  E,
E  E ,g [ (E E u  g )]( s )
 E  E ,g [ ' (E E u  g ) E ( s )]

 '( E  u  g )
 (s)
 | ( E )  0 [  ( ) ( E )  '( E u  g ) ] ' (E E u  g )  ( E )
E
  | ( E )  0  ' (E u  g )  ( ) ( s )
 E  [ ' (E u  g ) ( s )]
 E  [ (E u  g )]( s ).
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Proof Sketch
Part 2: DC implies Smooth Rule
• Consider a feasible set with smooth boundary at g.
• Now necessary for DC that the gradients of the
unconditional and conditional representations are
proportional on E at g.
• Considering a  with two-point support
qg
and some moderately messy algebra
g
shows the only reweighted Bayes’ rule
satisfying this proportionality is
the Smooth rule.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
DC updating of MEU preferences
•
Consider update rules that:
1. Leave u unchanged, and
2. Make Ec a null event.
Proposition (Prop. 1, Hanany and Klibanoff (2007)):
Such rules are dynamically consistent iff
arg min
pCE ,g ,B
 (u  g)dp Q
E, g ,B
( E)  
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Dynamic Ellsberg
example with our rule
 is uniform over   {( 13 , 92 , 94 ) , ( 13 , 94 , 92 )}
 ( x )  e
x
,  0
Satisfies modal Ellsberg choices (Jensen) :
V (1,0,0)   ( 13 )  V (0,1,0)  12 [ ( 92 )   ( 94 )]
and similarly,
V (0,1,1)  V (1,0,1) .
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Dynamic Ellsberg
example with our rule (cont.)
Assume g  (1,0,0)
(betting on Black).
Updating on {not Yellow}, the support of  E,g
is the two conditiona ls {( 53 , 52 ,0) , ( 73 , 74 ,0)} .
However,  E,g is distorted away from uniform.
Ellsberg choices are maintained (Jensen) :
VE , g (1,0,0)  VE , g (0,1,0)   ln( e
1
7
12
 7

5
12
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences

e5 )  0
Dynamic Ellsberg
example with our rule (cont.)
For the other pair of Ellsberg acts,
now assume g '  (0,1,1)
(betting against Black).
The support of  E,g is the same as for the first pair,
but the distortion away from uniform is different.
 E,g ' ( 53 , 52 ,0) 
1
( 6  )
7
1 5 e 35
while  E, g ( 53 , 52 ,0) 
,
1
6
7
1 5 e 35
.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Introduction (cont.)
• We begin by proposing an update rule for ambiguity
averse Smooth Ambiguity preferences and show it is the
unique dynamically consistent rule among a large class of
rules generalizing Bayes’ rule.
• We then provide a general result characterizing
dynamically consistent updating for any ambiguity averse
preferences and apply it to Smooth Ambiguity
preferences and to Variational preferences
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
3-Color Ellsberg Paradox
• Urn with 90 balls, exactly 30 are Black, no information about
the proportions of Red and Yellow.
• One ball drawn from Urn at random. Consider Preferences
over bets on the color of the drawn ball.
• Modal preferences:
Bet on Black > Bet on Red ~ Bet on Yellow
Bet against Black > Bet against Red ~ Bet against Yellow
• Reflects aversion to ambiguity about probabilities in this
context.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
3-Color Ellsberg Paradox
30 +
Black
?
Red
+
?
=
90
Yellow
Bet on Black
1
0
0
Bet on Red
0
1
0
Bet against Black
0
1
1
Bet against Red
1
0
1
Bet on Black > Bet on Red
Bet against Black > Bet against Red
• Conditional on ball drawn being {not Yellow}, a Bet on
Black is identical to a Bet against Red. Similarly, a Bet
on Red is identical to a Bet against Black.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
Comparison with Recursive
preferences
• As our rule is non-consequentialist, it is not
recursive.
• As our example showed, no recursive (or even
consequentialist) rule is capable of dynamic
Ellsberg behavior.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
DC updating of MEU preferences
Corollary (HK (2007)): Such rules are dynamically
consistent iff
For some r  Q E , g , B ,
rE  C E , g , B  q  E  (u  g )dq  (u  g )drE 




Proposition (HK (2007)): The unique ambiguity
maximizing dynamically consistent update rule is:
C E , g , B  q  E  (u  g )dq  arg min rQ E ,g ,B (u  g )drE 



E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences

Toward a General Result
Thus far, we have identified a relatively simple
rule for updating Smooth Ambiguity preferences
that results in dynamic consistency and other nice
properties.
Q: Is there a way to describe the set of "all"
dynamically consistent update rules for these
preferences? For other ambiguity averse
preference models (even ones with kinks)?
A: Yes!!
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
A General Result
Definition: If V is a concave function of utility-acts then its
superdifferential at x is:
∂V(x)≡{r∈Δ∣V(z)-V(x)≤ ∫(z-x)dr for all z}
Definition: For V, the measures supporting the conditional
indifference curve at h on E are:
GE,h(V) ≡ {q∈Δ(E)∣∃λ>0 such that q∈λ∂V(a) evaluated at a=u∘h}.
Definition: The measures supporting the conditional optimality of
h are:
QE,h,B ≡
{q∈Δ∣q(E)>0 and ∫(u∘h)dq ≥ ∫(u∘f)dq for all f∈B with f=h on Ec}.
Denote by QE,h,B(E) the set of Bayesian conditionals on E of measures
in QE,h,B.
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
A General Result
Proposition: Fix a preference relation, ≿E,g,B, on
Anscombe-Aumann acts, a vN-M utility u representing
≿E,g,B on constant acts, and an event E∈Σ.
If there exists a concave VE,g,B such that
1. f ≿E,g,B h if and only if VE,g,B (u∘f) ≥ VE,g,B (u∘h)
2. a(s) = b(s) for all s∈E implies VE,g,B (a) = VE,g,B (b), and
3. a(s) > b(s) for all s∈E implies VE,g,B (a) > VE,g,B (b),
Then,
[h ≿E,g,B f for all f∈B with f = h on Ec]

GE,h(VE,g,B) ∩ QE,h,B(E) ≠ .
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences
A General Result
•
This gives us a test for dynamic consistency of
an update rule for any concave model:
1. Calculate GE,g(VE,g,B)
2. Calculate QE,g,B(E)
3. See if they intersect.
•
What happens when we apply this to concave
Smooth Ambiguity preferences?
E. Hanany and P. Klibanoff ES-NASM 08
Updating Ambiguity Averse Preferences