INTERPRETATION OF UTILITY FUNCTIONS
A utility function is a relation that assigns numerical indicators of satisfaction (i.e. utilities) to
different possible outcomes. It is important to understand the meaning of these functions, hence we
need to interpret their representation.
The main goal of utility functions is to show the decision maker’s behaviour in relationship with the
choices he makes and the outcomes that derive from them. Specifically, by observing the shape of a
utility function, we can understand the decision maker’s attitude towards the risk.
Let’s start considering this kind of function:
u(x)
u(xa)
ū
x1 xc xa
x2
x
Utility functions having this concave shape provide evidence of risk aversion.
Risk aversion can be defined as the reluctance of a person to accept a bargain with an uncertain
payoff rather than another bargain with a more certain, but possibly lower, expected payoff.
Obviously, it can be easily demonstrated through the representation above: If we selected any two
points on this curve (in this case x1 and x2) and drew a straight line between them then it can be
seen that the curve would always be above the line. This means that the decision maker assigns a
higher utility to the expected value of the lottery rather than the expected utility of the lottery itself.
Put in other terms, the concavity of the function implies that the utility the decision maker can gain
from what is called the certainty equivalent (xc) is the same of the utility he can gain from the
expected value (xa). It is obvious that a sure outcome would always be preferred over a risky bet
having the same expected value.
Now, let’s consider this function:
u(x)
u(xa)
u(xc)
xa
x
Utility functions having this convex shape provide evidence of risk-seeking attitude.
Risk-seeking can be defined as the attitude of a person who has a preference for risk. A risk-seeker
person prefers to participate in a gamble, rather than receiving an expected value for sure. This kind
of behaviour means that this person assigns a lower utility to the expected value of the lottery
respect to the expected utility of the lottery itself.
Put in simple terms, a risk-seeker perceive a higher utility when he gain the desired outcome from a
gamble. Once again, we can see this attitude represented on the graph above: the utility of the
expected value of the lottery (ua) is greater than the utility of the certainty equivalent (uc) even
though we have the same expected value. For this reason, the convexity of the function implies a
risk-seeking attitude.
Finally, we can consider the last main case represented by this function:
u(x)
u(xa)
xa = xc
x
This type of functions provide evidence of risk-neutrality.
A person is risk-neutral, if the utility of the expected value of the lottery is equal to the expected
utility of the lottery itself.
In this case, it is very simple to understand that a person is indifferent between gaining a certain
outcome or participating in a lottery. In other words, a person is risk-neutral if the expected value of
the lottery is equal to the certainty equivalent. Hence, their utilities are the same.
References
Chiandotto B., Statistica per le decisioni (note didattiche), Firenze, 2006.
Goodwin P. & Wright G., Decision Analysis for Management Judgment, Third Edition, John Wiley & Sons, Ltd.
http://www.okpedia.it/neutralita_al_rischio
https://en.wikipedia.org/wiki/Risk_aversion