Von Neumann And Morgenstern Utility Function
Vishrut Goyal and Anuj S. Saxena
In last lecture we discussed the St. Petersburg Paradox, where although the expected return from the game is infinity, only a few gamblers are willing to invest large amount of money. Now we take a simplified version of the above paradox.
Consider a player being given two options : Take Rs.10 for sure or Rs.70 with probability 0.5. In this case, the player is likely to go for the second option.
Now consider the case when Rs 10 and Rs.70 were replaced by Rs. 10 million and Rs.70 million respectively. In this case, the user is likely to go for the first option. Why is it so? Why people may not want to maximize their expected gain?
Let be the initial amount of money with the user. Let be a utility function representing the user's preference relation. The expected utility in each of the options in the cases discussed above is summarized below :

Option 1 
Option 2 
Case A 


Case B 


Assume that the player has initially Rs. 10,000. i.e. .
Consider the case when . Now the expected utilities are given in the following table :

Option 1 
Option 2 
Case A 


Case B 


Here we see that the expected utility is more for option 2 in both the cases. Therefore players should prefer option 2 in both the cases
In earlier lecture we saw that if a utility function represents a preference relation R, then f(u()) also represents R, where f() is a monotonically increasing function. Therefore
should represent the same preference relation as . If we assume
, then the expected utilities are given in the following table :

Option 1 
Option 2 
Case A 


Case B 


Here we see that that expected utility is more for option 2 in case A and is more for option 1 in case B as one would generally expect in real life. Notice that
does not represent the same preference relation as . This can be explained using Von Neumann and Morgenstern utility function.
Let be a preference relation over a set of deterministic outcomes and
be a utility function representing . If we have probabilistic outcomes, then the preference relation should be defined over probabilistic outcomes
. Now the utility function has to be redefined so as to represent this new preference relation. Therefore, now
.
Notation : Let
, where is outcome in and is the probability of occurrence of s.t.
.
Von Neumannn and Morgenstern gave a set of rationality postulates, which define preference relation in probabilistic outcomes.
Consider any two outcomes (namely and ) from the set , with outcome being preferred over outcome . Let and . Consider two outcomes
and
in . A rational player should choose second outcome over the first. Formally :
s.t. and ,
If outcome is preferred over outcome and outcome is preferred over outcome , then there exists a probability p, such that the player is indifferent to the outcome
and . Formally :
If
, then
s.t.
Corollary : For , , defined as above, there exists a unique
s.t.
.
Proof : Suppose there exist
and (without loss of generality) s.t.
and
From first postulate, we know that:
which is a contradiction.
According to this postulate, if the player is indifferent between two outcomes say and , and in another outcome say , happens with probability , then the player remains indifferent if in is replaced by . To state this formally, label outcomes
in s.t.
. Consider an outcome in given as :
Let be the probability s.t.
.
If
, then
Von Neumann and Morgenstern utility function is defined over
. Let
is the most preferred outcome
and
is the least preferred outcome
. For each outcome , define

(1) 
From postulate 2, such a always exists and is unique.
Von Neumann and Morgenstern utility function for
is the expected value of the utility function u() as defined by equation 2. i.e.

(2) 
To check that the Von Neumann and Morgenstern utility function represents the preference relationship on , consider two outcomes and
in
.i.e.
and
By repeated application of postulate 3, we get :
and
From postulate 1, it is clear that given two outcomes and
in (as defined above), a rational player will prefer the one, which corresponds to higher expected value of the utility function, u().
Therefore,
represents a preference relationship on .
Vishrut Goyal
20021001