A utility function is a real-valued function that measures the "goodness" of an option to a decision maker, in the following sense:
- If there is no uncertainty about aspects of the world relating to preferences between Options A and B, Option A is preferred to Option B if and only if the utility of A is larger than the utility of B;
- If there is uncertainty, then Option A is preferred to Option B if and only if the mathematical expectation of the utility of Option A is greater than the mathematical expectation of Option B, where the expectation is with respect to the decision maker's probabilities for the relevant uncertainties.
A single attribute utility function is a numerical function that measures the "goodness" of a single attribute of value. Generally, utility functions are assessed by defining single attribute utility functions for each of the attributes and combining them into a multi-attribute utility function.
A single attribute utility function is often assessed by defining a measurable value function for the attribute. A measurable value function is monotonically related to utility. It satisfies the property that differences in the measurable value function correspond to differences in value to the decision maker. For example, if Options A, B, and C have values 0.2, 0.4,and 0.6, then the degree to which B is preferred to A is the same as the degree to which C is preferred to B. If uncertainty is important, then measurable value functions can be transformed into utility functions by assessing the decision maker's risk attitude (e.g., risk seeking or risk averse).