What is utility?

Decisions in connection with technical systems can include the following attributes (i.e., the quantities that need to be optimized)

• safety (or more generally human health), 
• environmental considerations,
• aesthetic appeal
• or monetary cost. 

Considering the different attributes separately will lead to a multitude of optimal decision alternatives. For example, monetary cost and safety are usually inversely related and, thus, a trade-off between monetary cost and safety must be made in the decision process.

Definition of utility

Therefore, a metric is needed that allows a comparison of the available decision alternatives quantitatively and measures the optimality (i.e., the desirability) of a decision. This metric should consistently reflect the preferences of the decision maker and is referred to as (cardinal) utility. The utility function $u(\mathbf{y})$ is expressed as a function of the attributes $\mathbf{y}$.

Illustrative example: In the simplest case, we have just a single attribute. Let it be money. Furthermore, let the utility increase linearly with an increase in monetary gain (which is not always the case); i.e., utility is proportional to negative cost. Under these conditions, we will prefer the decision alternative that has the largest monetary gain (i.e., the smallest cost). ■

The utility function is linked to the decision-maker

Note that different decision-makers can prefer differing utility functions.

Illustrative example: Have you ever wondered why on the one hand an insurance company decides to sell a policy and on the other hand the insurance holder decides to accept the policy? Shouldn't the decision be sub-optimal for either the insurance company or the insurance holder? The truth is that insurance can only makes sense (for both parties) if their utility functions differ. The utility function of the insurance company is usually linear with respect to the monetary gain, whereas the utility function of the insurance holder is a concave function with respect to monetary gain as he is risk averse. ■

Multi-attribute utility functions

The utility can be associated with a single attribute or with multiple attributes. In practice, it is often challenging to determine multi-attribute utility functions.

References

[Straub, 2020]  Straub, Daniel: Lecture Notes in Engineering Risk Analysis. Technische Universität München, 2020.