1. Deriving Bi-Symmetry Theorems from Tradeoff-Consistency Theorems
July 7, 2003; Peter P. Wakker (& Veronika Köbberling)
comments wegdoen
Mijn commentaren voor nonspecialists: see other files.
For specialists:
Dit vergelijkt TO-consistency en bisymmetry (1) intuitief. (2) wiskundig.
Explain immediately here that the results will be more general in a structural sense, but also in a logical sense.

2. {1,…,n}: states (of nature)2
{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

3.   ,…, ,…, ~ ,…, ~ Additivity: (for X = )3
Additivity: (for X = )
Anscombe & Aumann (1963) results if + is midpoint operation:
mixture-independence  x1 xi xn
,…, y1 yi yn 
,…, x1 xi xn y1 yi yn c1 + ci cn  c1 + ci cn
Transformation 1 of additivity
For all i:
Transformation 2 (under regularity): ci + 
,…, x1 xi xn y1 yi yn  ~ ~ 4 5

4.  ,…, ,…, Tranformation 3: For all i, ~ x1  xn y1  yn ~ x1  xn y1 4
Tranformation 3:
For all i, ~ x1  xn
,…, y1  yn  ~ x1  xn
,…, y1  yn
whenever  –  =  –  .

5.  nx = (x1,..,xn-1,) e.g. 1x = (,x2,..,xn), ix ~ iy5
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,
Bij generalization nog niet opbrengen dat U non-observable, dat komt op p. 7.
ix ~ iy 
ix ~ iy
whenever  –  =  –  .
U( )
utility
Definition. Subjective expected value holds if there exist p1,…,pn such that
(x1,…,xn)  p1x pnxn represents preferences.
, U
U(x1)
U(xn) 6 3 7

6. Theorem. The following two statements are equivalent: utility6
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;
(d) additivity.
tradeoff consistency .
(or mixture-independence).
This is my interpretation of de Finetti's book making theorem.
Virtually identical to Anscombe & Aumann.
now without linear utility/ extraneous addition
now without linear utility/ extraneous mixing 5
Or, this is Savage for finite state spaces. 10

7.  ix ~ iy As written before: For all i, ix ~ iy7
ix ~ iy
As written before:
For all i,
whenever  –  =  – . 
ix ~ iy
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!

8.  ix ~ iy Lemma. Under SEU,  holds. and ix ~ iy (under SEU)8
ix ~ iy
and
ix ~ iy 
(under SEU)
U() – U() = U() – U().
Lemma. Under SEU,  holds.

9.  ix ~ iy jv ~ jw ix ~ iy jv ~ jw9
Generalized additivity (tradeoff consistency)
ix ~ iy
and
jv ~ jw
This is used to reveal the U-differences equality. This can next be used to predict preferences à la de Finetti.
So this is what we do. We first observe two preferences to infer, “endogeneously,” the utility-difference equality. Next we use this to predict other preferences just as in de Finetti’s model.
and
ix ~ iy 
jv ~ jw
Lemma. SEU implies tradeoff consistency.
Proof. Otherwise, inconsistent equalities of utility differences would result. 6

10. and
ix ~ iy
ix ~ iy
jv ~ jw 
jv ~ jw
Generalized additivity (tradeoff consistency) was 10
Say that nice interpretation is desirable.
Question: Interpret how? What is intuition?
Endogenized addition operation through
U() + U() = U() + U()?
Improvement-comparison through
U() – U() = U() – U()?
I preferred another interpretation:
Interpret as improvement-comparison
U() – U() = U() – U().
Say that improvement-comparison should be consistent.
Say that addition operation by arbitrarily taking a  where U is zero. Require consistency of 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.)

11. ix ~ iy ix ~ iy  ~* . Notation: If  x,y, and nonnull i, s.t.11
Notation:
& If
ix ~ iy
ix ~ iy
 x,y, and nonnull i, s.t.
CEU is characterized by requiring these to be comonotonic.
Herhalen dat TO consistency SEU geeft, dus dat voor Savage voor finite state space voor uw studenten niet meer nodig is dan bovenstaande.
Zeggen dat al het voorgaande vergeten kan worden. Dit is de essence.
Easiest to understand and teach to students. Clear intuition (tradeoff, regret).
DERIVED CONCEPT …
Easiest to test in experiments. Has been used in experiments to measure utility.
then
 ~* .
CPT: these should also be cosigned.
Tradeoff consistency:
No inconsistencies in ~* elicitations.
not  ~*  and ' ~*  for '.
In words. Tradeoff consistency holds if:
improving an outcome in a ~* relationship breaks the relationship.
Adaptations, besides subjective expected utility:
Comonotonicity/Choquet expected utility: done.
Sign-dependence/prospect theory: done!

12. Alternative axioms, more commonly used, for jU(xj).12
Now for something else!
Alternative axioms, more commonly used, for jU(xj).
Based on bisymmetry.
Start with n=2.

13. F( ) = F(F(x1,y1),F(x2,y2)). = ( )13
A digression on functional equations. Let F: 2  
F( )
F(x1,x2) ,
F(y1,y2)
= F(F(x1,y1),F(x2,y2)).
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: x1 x2
CE(x1,x2) = x1 x2 x1 y1 x2 y2
Zeggen dat even alleen reele getallen, voor gemak.
s1: rain; s2: no rain
Aan eind gaan discussieren over intuitie. Zeggen: Waarom zou dit leuk zijn, of, bijv., normatief, als je iemand wilt overtuigen dat SEU normatief, waarom dan hieraan te voldoen?
Herhaalde gebeurtenissen …
Zelfde CE substitution. Informationally independent. Misschien te verkopen bij known probs. Anders moeilijk hoor!
Bisymmetry for CE: = y1 y2
( )
CE(y1,y2)
CE(x1,x2) , CE

14. ~ = right EU(left) = EU: left ~ right, i.e. bisymmetry is implied.
skippen 14 x1 x2 y1 y2 x1 y1 = ~ x2 y2
right
EU(left) =
1U(Uinv(1U(x1)+2U(x2))) + 2U(Uinv(1U(y1)+2U(y2))) =
1(1U(x1)+2U(x2)) + 2(1U(y1)+2U(y2)) =
12U(x1)+12U(x2)) + 21U(y1)+22U(y2)) y1 x2 y1 x2 y1 x2
Na laatste regel komt er nog dat gelijk-teken boven vervangen wordt door indifferentie-teken!
EU: left ~ right, i.e. bisymmetry is implied.
Bisymmetry for general outcomes:

15. Now notation etc. for general n.15
Now notation etc. for general n. fi fn f1 .
= CE(f1,…,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. ~ ( ) Assume event A as above. = CE16
Assume event A as above. fi f1 fn =
CE(f1,…,fn)
CE(g1,…,gn) ,
Ac: CE ) A: ( g1 gi gn ~ f1 g1 . gi fi
= CE(CE(A:f1,Ac:g1), …, CE(A:fn,Ac:gn))
Bij uitleg zeggen dat twee acts have same normal form. . fn gn
Multisymmetry with respect to A.
Hier: In the first race a French horse will win. In the repeated race horse 1 wins. Later komt de repeated race eerst.
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:17
Rewriting this separability, with event A fixed as before: c1 ci f1 fi fn cn .
f ´1
f ´i
f ´n ci c1 cn . fi f1 fn
f ´i
f ´1
f ´n   
Same for Ac.
Act-independence (Gul 1992)

18. Theorem. Assume connected separable product topology.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.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. Then tradeoff consistency holds.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! 

21.  ~ ~ ~ ~ ~ ~ ~ ~ ~ Similarly, ~  ~*  ’ ~*   ’ ~ must be.
f´i
f´n
g´n
f´1
g´i  f1 fi fn
g´1 g1 gi gn .
and 21 ~ z a c x b y
 ~*  h h h h h f1 g1 f1 g1 f1 . . f1 g1 g1 f1 g1 h h ~  ~    .  . .   fn gn . . h h h h h  ~  h    fn gn    ~ ~
and
and    g1 f1      . . . g1 f1 . .
Similarly,
’ ~*    ~  h  '  . .    ~  ~ h h h   . . gn fn . . fn gn fn gn fn gn fn gn
’ ~ must be.
Tradeoff consistency holds.     

22. 22
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).

23. - Gul (1992, JET, “Savage … Finite States”) Pfanzagl (1959); 23
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).

24. Need to consider only one mixing event A, not all. 24
Generalizations of multi-symmetry-based results by means of the above derivation:
Need to consider only one mixing event A, not all.
Don’t need topological separability, can do, even more generally, for solvability.
Need only indifferences, not preferences, in preference conditions.

25. Arguments in favor of tradeoff techniques: 25
Arguments in favor of tradeoff techniques:
More intuitive and simple (? matter of taste, Luce disagrees …) Tradeoffs play role in decisions, not meta …
More general mathematically. (Can be used for prospect theory.)
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.)
(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.