AI Glossary / Graphs and symbolic AI / Intermediate
Ordinal utility function
Definition
An ordinal utility function assigns numbers to alternatives so that their ordering represents a decision-maker's preferences. A higher number means the alternative is preferred, while equal numbers represent indifference within the model. The numerical gaps and ratios do not, by themselves, measure how much more an alternative is valued. Its essential information is the ranking.
What it means
Suppose a person prefers option A to B and B to C. Scores of 3, 2, and 1 can represent that order, but so can 100, 10, and 0. The second scale does not show stronger preferences merely because its numerical gaps are larger.
In AI, this distinction matters when a ranking or preference model supplies scores. Those scores may be useful for choosing among known alternatives without supporting arithmetic claims about satisfaction, trade-offs between people, or decisions involving uncertain outcomes. The intended interpretation needs to be stated before the numbers are reused elsewhere.
How it works
Start with a preference relation and ask whether a numerical function can represent it consistently. For a finite set, a complete and transitive preference relation can be represented with ordinal utility values. Tied alternatives receive the same value, and preferred alternatives receive higher values.
Any strictly increasing transformation preserves that ordering. The transformed values therefore describe the same ordinal preferences, even if their gaps change greatly.
Expected-utility comparisons require additional structure. Applying an arbitrary increasing transformation to outcome utilities and then averaging can change the ranking of lotteries. An expected-utility representation is preserved by positive affine transformations, meaning multiplication by a positive constant followed by adding a constant. This is a narrower freedom than ordinal ranking alone permits.
Three practical examples
- Personal recommendations: a user ranks three learning activities without claiming that the first is twice as enjoyable as the second.
- Interface choices: a design team records an individual's preference order among layouts before selecting from the available set.
- Workflow selection: an assistant uses a stated ranking of complete, certain outcomes to choose a preferred feasible option, while avoiding unsupported calculations about preference intensity.
Think of it this way
Think of positions in a race. First, second, and third tell you who finished ahead, but the positions alone do not reveal whether the gaps were fractions of a second or several minutes. Ordinal utility similarly preserves order without supplying meaningful numerical distances. Unlike a race, a preference model may include ties and does not necessarily reflect a physical measurement such as finishing time.
Worked example
Consider outcomes A, B, and C, ranked A above B above C. Assign ordinal utilities 3, 2, and 1. Cubing these positive values gives 27, 8, and 1. Both assignments preserve every pairwise ranking.
On the first scale, A exceeds B by one and B exceeds C by one. On the second, the gaps are nineteen and seven. Consequently, equal gaps on the first scale cannot be interpreted as an established equality of preference intensity. The same preferences admit unequal gaps after transformation.
Now compare a certain B with a lottery that gives A with probability 0.4 and C with probability 0.6. Averaging the first numbers gives the lottery 0.4 × 3 + 0.6 × 1 = 1.8, below B's 2. Averaging the cubed numbers gives 0.4 × 27 + 0.6 × 1 = 11.4, above B's 8.
The certain-outcome ranking stayed unchanged, yet the arithmetic lottery comparison reversed. This proves that the ordinal ranking alone does not determine which of these uncertain options the person prefers. It does not prove that either average is the person's actual expected-utility assessment; that requires preferences over uncertain choices and an appropriate representation.
Illustrative case study
Fictional case: Meadow Learning asks a tester to rank three course formats. The tester prefers live workshops, then recorded lessons, then text summaries. A product analyst records these as 3, 2, and 1.
A later report describes workshops as “three times as valuable” as text summaries. A reviewer removes that statement because the original exercise elicited only an order. The team also declines to infer the tester's preference between a guaranteed recorded lesson and an uncertain chance of the other formats.
The case illustrates a measurement correction. It makes no claim about real users, course effectiveness, or commercial outcomes.
Common misconceptions and mistakes
Do not interpret an ordinal utility of ten as twice the satisfaction of five. Ratios are not established by the ranking representation.
Do not assume a zero value means absence of all value or enjoyment. Its numerical location can change while the preference order remains identical.
Avoid combining different people's arbitrary scales as if their units were automatically comparable. Also distinguish an ordinal utility model from an ordinal label such as a star rating: the label alone does not establish the full preference assumptions. Incomplete or cyclic preferences require attention rather than being silently forced into a faithful single-number ranking.
Business use
Ordinal information can still support useful choices. If all feasible outcomes are certain and one is preferred to every other, an order-preserving rescaling does not change that choice.
For business analysis, record whether a study measured order, intensity, or choices under uncertainty. In Meadow Learning's example, the ranking supports selecting the tester's favorite available format. It does not support a return-on-investment calculation or a claim about how much more the tester would pay. Matching the analysis to the elicited information prevents an attractive dashboard number from carrying more meaning than the evidence supports.
People also ask
Are ordinal utilities useless because their numbers are arbitrary?
No. They can represent preference order and support choices that depend only on that order.
Does a strictly increasing transformation change the preferred certain option?
No. It preserves the ranking, including indifference when equal original values remain equal.
Can ordinal utilities automatically be averaged over uncertain outcomes?
No. The worked example shows that different representations of the same certain-outcome ranking can reverse such a comparison.
Check your understanding
Do 3, 2, 1 and 27, 8, 1 represent different rankings of A, B, and C?
Answer: No. Both rank A above B above C. They differ numerically, but their ordinal information is the same.
Related terms
- Utility function
- Ordinal scale
- Expected utility
- Completeness
- preference relation
- indifference
- positive affine transformation
- transitivity
Learn next
create two different numerical representations of the same ranking and check which conclusions survive rescaling. Then study preferences over lotteries and the assumptions needed for expected utility before using ranking scores in uncertain decisions.
More in Graphs and symbolic AI
- Optimism under uncertainty - Optimism under uncertainty is an exploration principle that gives insufficiently understood actions the benefit of plausible favorable outcomes.
- Optimizer's curse - The optimizer's curse is the tendency for an alternative selected because it has the highest noisy value estimate to perform worse, on average, than that...
- Ordering constraints - Ordering constraints specify that one action or event must occur before another.
- Orientation - Orientation describes the direction in which something is aligned relative to a chosen reference.
- Origin function - An origin function, in the BLOG probabilistic modeling language, maps a generated object to an object that participated in its generation.
- Out-of-bag error - Out-of-bag error is an estimate of predictive error obtained by evaluating each training example using only ensemble members whose bootstrap training...
Sources
Primary sources: Alexander Wolitzky, MIT, “Choice, Preference, and Utility”: https://ocw.mit.edu/courses/14-121-microeconomic-theory-i-fall-2015/b89cf6eb8d2118d015d4e41278ede14a_MIT14_121F15_1S.pdf Russell and Norvig, utility-scale slides: https://aima.cs.berkeley.edu/4th-ed/slides-pdf/chapter16.pdf Relevant representation and transformation passages were read. Numerical comparisons and Meadow Learning are original illustrations.
Previous: Ordinal tree Next: Ordinal variable