Titel_SS06

A further analysis of the decision problem requires the numerical assessment of the
preferences of the decision maker. It is assumed that the decision maker prefers B to , C
to

A

A, and B to. This statement of preferences may be expressed by any function u such
that:

C

uB uC uA() () ()&& (3.1)

The task is to find a particular function u namely the utility function such that it is logically
consistent to decide between and by comparing a 1 a 2 uC
 with the expected value of the

utility of the action , namely: a 1

puA() (1 )()puB (3.2)

where p is the probability that the state of nature is : 1.

Assuming that and have been given appropriate values the question is - what

value should have in order to make the expected value a valid decision criterion? If the

probability

uA()

uC



uB()

p of : 1 being the state of nature is equal to 0 the decision maker would choose

over because she prefers

a 1

a 2 B to C. On the other hand if the probability of : 1 being the state

of nature is equal to 1 she would choose a 2 over. For a value of a 1 p somewhere between 0

and 1 the decision maker will be indifferent to choosing over. This value a 1 a 2 p* may be

determined and uC  is assigned as:

uC()pu A**() (1p uB)() (3.3)

From Equation (3.3) it is seen that uC
 will lie between uA() and uB() for all choices of

p* and therefore the utility function is consistent with the stated preferences. Furthermore it
is seen that the decision maker should choose the action a to only if the expected utility

given this action

12 a

Eua 1  is greater than Eua 2. This is realized by noting that for all p

greater than p* and with uC
 given by Equation (3.3) there is:

**

*

() () (1 )()

() (1 )() () (1 )()

( ) (( ) ( )) ( ) (( ) ( ))

uC pu A p uB

puApuBpuApuB

uB uA uB p uB uA uB p

&

 & 

 &





(3.4)

This means that if is properly assigned in consistency with the decision makers stated

preferences i.e.

uC



B preferred to preferred to and the indifference probability C A p*, the

ranking of the expected values of the utility determines the ranking of actions.