1. Theoretical Background of the Utility Measurement Method of this Course
Peter P. Wakker
Remove yellow comments before presentation.
ALT-View-O
We introduce axiomatizations of rank-dependence, and rank- and sign-dependence, that are
More general;
simpler
more intuitive.
Convince you that RDU and PT are natural models, naturally following from intuitions.

2. Example 1: weather tomorrow; 2
{E1,…,En}: events partitioning universal event; suppressed from now on.
Example 1: weather tomorrow;
Example 2: horse race, exactly one horse will win. n horses participate.
Event j: horse j will win.
: outcome set, money;
x = (x1,…,xn): prospect, yields €xj if Ej (horse j wins), j = 1,…,n.
(10,–1, …,–1): get €10 if E1 (horse 1 wins), lose €1 otherwise.
: preference relation over prospects.

3. Notation: Eix is (x with xi replaced by ),3
Notation: Eix is (x with xi replaced by ),
For example
10E1x = (10,x2,..,xn),
Enx = (x1,..,xn-1,)
Monotonicity:     Eix  Eix;

4. U: X: utility function; pj : (subj) prob. of horse j. 4
Expected utility
(x1,...,xn)   pj U(xj) n
j=1
U: X: utility function;
pj : (subj) prob. of horse j.
pj  0; pj = 1.

5. Then 2  1 1  0 (same with 3, 4) .5
Example of our technique:
Six events {E1,…,E6}. Suppose
(0, ,0,…, ) (0, ,0,…, )
190
800 ~
100
900
(0, ,0,…, ) (0, ,0,…, )
610
800 ~
500
900
Then we write 610  500 ~ 190 100 . t
Immediately add interpretation of extra happiness. Otherwise it is too dry and abstract, and not normative. Use term improvement iso tradeoff.
Adapt below to story for students did in class.
Our experiment (Ej: candidate j wins)
( , 1) (0, 8)
Voor ~t notatie het in woorden zeggen.
Opmerken dat ~t alleen within-person comparison doet. 1 ~
( , 1) (1, 8) 2 ~
Then 2  1 1  0 (same with 3, 4) . ~ t

6. Eix ~ Eiy then    ~    If6 If
there exist prospects x,y, and nonnull Ei with:
Eix ~ Eiy
(Ei )U()+ji(Ej )U(xj) =
(Ei )U()+ji (Ej )U(yj)
and Eix ~ Eiy
(Ei )U()+ji (Ej )U(xj) =
(Ei )U()+ji (Ej )U(yj)
then (Ei )U() - (Ei )U() = (Ei )U() - (Ei )U()
then    ~    t
U() – U() = U() – U()
Lemma. Under (subj) expected utility,
 ~   U() – U() = U() – U() . t

7.  U() – U() = U() – U() . (2) ´ ~t  7
A natural condition: EU
(1)  ~t 
 U() – U() = U() – U() .
(2) ´ ~t 
 U(´) – U() = U() – U() .
 ´ ~ .
U(´) = U()
Improving an outcome in any ~t relationship breaks the relationship: tradeoff consistency.

8.  ~ ´ Ei x ~ Ei y Ei x ~ Ei y  Ej f ~ Ej g (1)  ~t 8
(1)  ~t 
 U() – U() = U() – U() .
(2) ´ ~t 
 U(´) – U() = U() – U() .
 ´ ~ .
U(´) = U()
Improving an outcome in any ~t relationship breaks the relationship:
tradeoff consistency.
Skip part below.
In preferences: There are no nonnull Ei, x, y, and nonnull Ej, f, g with:
Ei x ~ Ei y
Ei x ~ Ei y
(1)
Voor de verbale expressie in regel vier het in woorden zeggen over  .
Voor “In other words” publiek voorbereiden door de zeggen dat ~t geen primitieve is.
 ~ ´ /
´Ej f ~ Ej g
 Ej f ~ Ej g
(2)

9. The following two statements are equivalent: 9
Theorem.
The following two statements are equivalent:
(i) (cont. subj) expected utility.
(ii) four conditions:
(a) weak ordering;
(b) monotonicity;
(c) continuity;
(d) tradeoff consistency.
Compare to vNM independence.
Is Savage (1954) for (possibly) finite state spaces.
End of lecture.