Deriving Bi-Symmetry
Theorems from TradeoffConsistency Theorems
July 7, 2003; Peter P. Wakker (& Veronika
Köbberling)
comments
wegdoen
{1,…,n}: states (of nature)
Example: horse race, exactly one horse
will win. n horses participate.
State j: horse j will win.
X: outcome set; general (finite, infinite)
x = (x1,…,xn): act,
yields xj if horse j wins,
j = 1,…,n.
Xn: set of acts
 on Xn: preference relation of decision maker
2
3
Additivity: (for X = )
x1 ,…, xi ,…, xn
 y1 ,…, yi ,…, yn
Anscombe &
Aumann

(1963) results
x1 ,…, xi ,…, xn
y1 ,…, yi ,…, yn if + is
 +
+
+
+
+
+
midpoint
ci
c1
cn
ci
c1
cn
operation:
mixtureTransformation 1 of additivity Transformation
2
independence
For all i:
(under regularity):
x1 ,…, xi ,…, xn ~ y1 ,…, yi ,…, yn

x1 ,…, xi ,…, xn
+
ci
y1 ,…, yi ,…, yn
~
+
ci
4
5
4
Tranformation 3:
For all i,
x1 ,…,  ,…, xn ~

x1 ,…,  ,…, xn
~
y1 ,…,  ,…, yn
y1 ,…,  ,…, yn
whenever  –  =  –  .
5
Notation: ix is (x with xi replaced by ),
e.g. 1x = (,x2,..,xn), nx = (x1,..,xn-1,)
Generalization
Transformation 4 of additivity:
For all i,
ix ~ iy

ix ~ iy
Bij generalization
nog niet
opbrengen dat U
non-observable,
dat komt op p. 7.
whenever U( ) – U() = U( ) – U().
utility
Definition. Subjective expected value holds if there
exist p1,…,pn , U such that
(x1,…,xn)  p1x1 + ... + pnxn represents
U(xn)
preferences. U(x1)
6
3
7
6
Theorem.
The following two statements are equivalent:
utility
(i) Subjective expected value holds.
with U increasing and continuous.
(ii) Four conditions hold:
(a) Weak ordering;
(b) monotonicity;
(c) continuity;
tradeoff
consistency
(d) additivity..(or
mixture-independence).
This is my interpretation of de Finetti's book making
theorem. Virtually identical to Anscombe & Aumann.
now without linear utility/ now without linear utility/
extraneous addition
extraneous mixing
Or, this is Savage for finite state spaces.
5
10
7
As written before:
For all i,
ix ~ iy

ix ~ iy
whenever U( ) – U( ) = U( ) – U(.)
Problem:
How observe "="?
U is not directly observable!
How "endogenize ="?
Answer: Is staring at your face!
The old "reversal-trick" of revealed preference.
If you can derive a preference from a model,
then you can derive the model from the preference!
ix ~ iy
and
ix ~ iy

(under SEU)
U() – U() = U() – U().
Lemma. Under SEU,  holds.
8
9
Generalized additivity (tradeoff consistency)
ix ~ iy and jv ~ jw
So this is what we do. We
This
is used to reveal
first observe two preferences
and
to infer,
“endogeneously,” the
the
U-differences
utility-difference equality.
ix ~ iy
Next we use this
to predict
equality.
This
can next
other preferences just as in
to predict
Finetti’s model.
jv ~ jw bedeused

preferences à la de
Finetti.
Lemma. SEU implies tradeoff consistency.
Proof. Otherwise, inconsistent equalities of utility
differences would result.
6
Generalized additivity (tradeoff consistency) was
ix ~ iy and jv ~ jw
Say that nice
and
interpretation is
ix ~ iy
desirable.

jv ~ jw
10
Question: Interpret how? What is intuition?
Say that improvementEndogenized addition operation
through
comparison should be
U() + U() = U() + U()? consistent. Say that addition
operation by arbitrarily
Improvement-comparison through
taking a  where U is zero.
Require consistency of
U() – U() = U() – U()?
the operation.
Above things are meta in sense of comparing different decision
situations. Go on in minds of us researchers trying to model
different decisions. Last thing has meaning in single decision
situation. it goes on in mind of decision maker. It's their
influence. Rational decisions weigh pros against cons.
(Irrational version: regret theory.)
I preferred another interpretation:
Interpret as improvement-comparison
U() – U() = U() – U().
Notation: If  x,y, and nonnull i, s.t.
Herhalen dat TO
ix ~ iy &
CEU is characterized
consistency SEU geeft,
 i x ~ i y
then  ~* .
11
by requiring these to be
dus dat voor Savage voor
comonotonic.
finite state space voor uw
studenten
niet should
meer nodig
CPT:
these
is dan
also
bebovenstaande.
cosigned.
Tradeoff consistency:
No inconsistencies in ~* elicitations.
not  ~*  and ' ~*  for '.
Zeggen dat al het
voorgaande vergeten kan
worden. Dit is de essence.
Easiest to understand and
teach to students. Clear
intuition (tradeoff, regret).
In words. Tradeoff consistency holds if:DERIVED CONCEPT …
Easiest to test in
improving an outcome in a ~* relationship
experiments. Has been
breaks the relationship.
used in experiments to
measure utility.
Adaptations, besides subjective expected utility:
Comonotonicity/Choquet expected utility: done.
Sign-dependence/prospect theory: done!
12
Now for something else!
Alternative axioms, more commonly used,
for jU(xj).
Based on bisymmetry.
Start with n=2.
F: 2 
A digression on functional equations. Let
F(x1,x2),F(y1,y2)) = F(F(x ,y ),F(x ,y )).
F(
1 1
2 2
Reflexive solution:
F(x1,x2) = Uinv(1U(x1) + 2U(x2)).
For DUU: F is Certainty equivalent (CE) under EU!
Enough about F and functional equations.
Back to DUU, for now with n=2.
Alternative notation for CE, as an operation:
CE(x1,x2) =
x1
x2
Bisymmetry for CE:
x1
x1
x2
y1
y1
CE (CE(x1,x2) , CE(y1,y2))
y2
=
x2
y2
13
skippen
14
x1
x1
x2
y1
y1
y2
=
~
x2
y2
right
EU(left) =
y1
x2
inv
1U(
(1U(x1)+2U(x2))) + 2U(U (1U(y1)+2U(y2))) =
x2
y1
1(1U(x1)+2U(x2)) + 2(1U(y1)+2U(y2)) =
y1
x2
12U(x1)+12U(x2)) + 21U(y1)+22U(y2))
Uinv
EU: left ~ right, i.e. bisymmetry is implied.
Bisymmetry for general outcomes:
15
Now notation etc. for general n.
f1
..
.
fi
..
.
= CE(f1,…,fn)
fn
For general n, and a fixed event A,
we can still define a binary certainty equivalent
operation as follows:
x1
x2
= CE(A:x1, Ac:x2)
16
..
.
..
.
f1
g1
fi
gi
fn
gn
Assume event A as above.
= CE(A: CE(f1,…,fn) , Ac: CE(g1,…,gn) )
~
f1
fi
fn
g1
gi
gn
= CE(CE(A:f1,Ac:g1), …, CE(A:fn,Ac:gn))
Multisymmetry with respect to A.
Bij uitleg zeggen
dat twee acts
have same
normal form.
Multisymmetry implies that,
in CE(CE(A:f1,Ac:g1), …, CE(A:fn,Ac:gn)),
(f1, …,fn) and (g1, …,gn) are separable.
Publiek erop
wijzen dat de CE
na 2e = teken
komt.
17
Rewriting this separability, with
event A fixed as before:
f
1
f1
fi
fn

f´1
f´i
f´n
..
.

c1
fi
ci
.
..
.

..
fn
cn
Same for Ac.
Act-independence (Gul 1992)
..
.
f´1
c1
f´i
ci
f´n
cn
18
Theorem. Assume connected separable
product topology.
The following two statements are equivalent:
(i) Subjective expected utility holds with U continuous.
(ii) Four conditions hold:
(a) Weak ordering;
(b) monotonicity;
(c) continuity;
(d) act-independence w.r.t. all nontrivial
events A.
or multi-symmetry.
19
Lemma. Assume weak ordering,
monotonicity, a CE for each act.
Multi-symmetry for some nontrivial event

act-independence for some nontrivial event

tradeoff consistency.
Proof.
20
Preparatory lemma. Assume that X is a convex
set, and assume weak ordering, monotonicities,
and mixture independence:
f ~ f´  ½f + ½g ~ ½f´ + ½g.
Then tradeoff consistency holds.
Proof. Assume  ~* , i.e.
ix ~ iy and ix ~ iy.
Twofold mixture-independence:
½ix + ½iy ~ ½iy + ½iy ~ ½iy + ½ix.
Monotonicity: ½ + ½ ~ ½ + ½.
Likewise, ´ ~*   ½´ + ½ ~ ½ + ½.
Monotonicity:  ~ ´. Tradeoff consistency follows!

a
b
b
c
x
b
x
b
y
~
z
~
~
b
y
~
y
z
b
a
a
y
b
y
c
x
c
~
~
y
y
z
f1
~
fi
 ~* 

~
fn

gn
g1
.
. ~
.
g. n
.
.
f1
.
.
..
fn
.
.
f´i
g1
g1
.
. f´
.
i
. gi
g´1
.
. f´
i
.
. g´i

~
g´n
gn
h
h
h
f1
.
.h
.
g1
.
.h
.
f1
g1
g1
f1
f1
g1
. 
.
.
h
. 
.
.
h
. 
.
.
h
.
.
.
.
.
.
~
.
.
.
h

h
fn
gn
g1
f1




.
. gn
.

.
.
.
~
~

.
.
.



.
.
.

~
.
.
.
~
.
.
h



~





Similarly,
h
h
h
fn
gn
gn
fn
fn
gn



~
.
.
f´n
’ ~*  
f´n
g´n
gn
h


.
.f
n
.

gi
fn
h

fi
g´i
h
and
and
f´1
g´1
g1
gn
g1
f´1
and
gi
f1
f1
f´n
fn
21
f´1
h
h
'
~




’ ~ must be.
Tradeoff consistency holds.
The preceding results can be derived under
comonotonicity, and also when restricted to
binary acts, without any complication.
The following results, for these contexts,
now become corollaries of the above
theorems.
For risk:
- Quiggin (1982)
- Chew (1989 unpublished)
- Chew & Epstein (1989, JET, regarding
their RDU results).
22
For uncertainty
Nakamura (1990, 1992).
- Gul (1992, JET, “Savage … Finite States”)
- Pfanzagl (1959);
- Chew & Karni (1994);
- Ghirardato & Marinacci (2002, MOR);
- Ghirardato, Maccheroni, Marinacci, & Siniscalchi,
Proposition 4 (Econometrica, forthcoming).
Only paper with rich outcomes that is not a direct
corollary is, to the best of my knowledge,
Chateauneuf (1999, JME).
23
24
Generalizations of multi-symmetry-based results
by means of the above derivation:
(1) Need to consider only one mixing event A, not
all.
(2) Don’t need topological separability, can do,
even more generally, for solvability.
(3) Need only indifferences, not preferences, in
preference conditions.
25
Arguments in favor of tradeoff techniques:
(1) More intuitive and simple
(? matter of taste, Luce disagrees …)
Tradeoffs play role in decisions, not meta …
(2) More general mathematically.
(Can be used for prospect theory.)
(3) Due to many certainty equivalents, other
conditions are harder to test.
(Many empirical studies use tradeoff technique,
none that I know of uses other techniques.)
(4) (Ad 1): Intuition of other techniques based on
(hypothetical) multistage decisions with folding
back. Folding back is controversial, and
“subjective independence” is hard to justify.