Does utility equal money?

Not generally. A person's utility for money can be nonlinear, reflecting preferences about risk and other circumstances.

Can normalized utility lie outside zero and one?

If new outcomes extend beyond the selected reference anchors, the same affine rule can produce values outside that range.

What if the best and worst utilities are equal?

The proposed formula divides by zero; distinct utility anchors are required.

Are other normalization conventions possible?

Yes. State the convention and check which comparisons it preserves in the relevant framework.

AI Glossary / Statistics and probability / Intermediate

Normalized utilities

Definition

Normalized utilities are utility values placed on a specified reference scale, commonly zero for a chosen worst outcome and one for a chosen best outcome. In expected-utility decision theory, a positive affine transformation changes the numerical scale while preserving preferences over lotteries. Normalization makes the representation convenient; it does not turn utility into a probability, establish agreement between people, or determine which outcomes should be preferred.

What it means

A utility function represents preferences under a stated decision model. Its numerical origin and unit can be arbitrary. Assigning common anchors helps communicate the scale and calculate expected utilities consistently.

For bounded reference outcomes with utilities uWorst and uBest, where uBest exceeds uWorst, one normalization is u'(o) = [u(o) − uWorst]/[uBest − uWorst]. Outcomes between the anchors receive values between zero and one. The anchor choices and the preference interpretation must be documented; a label such as “0.8 utility” has little meaning without them.

How it works

Define the outcomes and whose preferences the utility function represents. Identify distinct reference outcomes and either rescale an existing expected-utility function or elicit preferences using a method consistent with the model.

One theoretical elicitation asks for the probability p at which a person is indifferent between an outcome and a lottery giving the best reference outcome with probability p and the worst with probability 1 − p. With utilities anchored at one and zero, the outcome receives utility p under the expected-utility assumptions.

For any positive a and constant b, replacing u with au + b preserves expected-utility comparisons because expected transformed utility is a times expected utility plus b. Arbitrary increasing nonlinear transformations preserve the ordering of certain outcomes but need not preserve choices between lotteries. That distinction separates cardinal expected-utility scaling from a simple ordinal ranking.

Three practical examples

A decision-support prototype rescales an already specified utility function to zero and one for clearer presentation.
An AI planning exercise compares actions using expected utility while retaining the same outcome anchors throughout.
A workshop checks whether two teams' apparent scoring disagreement is merely a scale difference or a real disagreement about tradeoffs.

Think of it this way

Think of expressing temperature in two different units. A consistent change of origin and unit can preserve the underlying comparisons while changing every displayed number. Utility normalization similarly changes representation, but it cannot settle disagreements about which outcomes are desirable or how uncertainty should be valued.

Worked example

Consider an original decision model with three outcome utilities: 20 for the worst reference outcome, 60 for an intermediate outcome, and 100 for the best. Applying u' = (u − 20)/80 gives normalized utilities 0, 0.5, and 1.

Action A guarantees the intermediate outcome. Its expected utility is 60 on the original scale and 0.5 on the normalized scale. Action B gives the best outcome with probability 0.6 and the worst with probability 0.4. Its original expected utility is 0.6 × 100 + 0.4 × 20 = 68; its normalized expected utility is 0.6. Both scales prefer B to A.

Now compare A with a different action C that gives the best outcome with probability 0.4 and the worst otherwise. A has normalized expected utility 0.5, exceeding C's 0.4. If someone incorrectly squares each normalized outcome utility, A's value becomes 0.25 while C's expected value remains 0.4, reversing the preference. Squaring preserves the order of these certain outcomes but not their expected-utility comparisons. The utilities and probabilities are invented to demonstrate scale properties, not to recommend an actual business action.

Illustrative case study

Fictional case study: Maple Planning builds an AI-supported option comparison. One analyst uses a utility scale from 20 to 100, while another presents its equivalent zero-to-one version. The resulting rankings of uncertain actions agree once the transformation is documented.

A stakeholder then proposes squaring the normalized scores to make attractive outcomes “stand out.” The reviewer explains that this changes the represented risk preferences rather than merely changing units. Maple separates presentation choices from substantive revisions to the utility model. The fictional case illustrates why a visually convenient transformation can change a decision rule.

Common misconceptions and mistakes

“A utility of 0.8 means an 80% chance of success.” Utility describes preference; probability describes uncertainty. They play different roles in expected utility.

“Any increasing transformation preserves expected-utility choices.” Only positive affine transformations preserve all such comparisons for a fixed expected-utility representation.

“Normalization makes different people's utilities directly comparable.” Common endpoints do not by themselves establish interpersonal comparability or fair aggregation.

“Normalized utilities must sum to one.” They are values assigned to outcomes, not necessarily a probability distribution over those outcomes.

Business use

Normalized utilities can make a decision model easier to review by showing its reference outcomes and tradeoffs on a familiar scale. Their usefulness depends on the quality of the preference model and the completeness of the outcomes considered.

Keep preference judgments distinct from predicted probabilities. If an AI system supplies outcome estimates, those estimates should not silently determine the utility scale as well. Test whether a recommendation changes under plausible alternative utilities, probabilities, or missing consequences. For several stakeholders or criteria, document the aggregation assumptions rather than assuming that putting everything between zero and one resolves competing values.

Check your understanding

Why does adding a constant to every outcome utility preserve a fixed lottery comparison?

Answer: Lottery probabilities sum to one, so the same constant is added to each action's expected utility. Their ordering is unchanged. A positive multiplier likewise preserves that ordering.

Related terms

Learn next

Compare ordinal outcome rankings with preferences over lotteries. Then verify expected-utility invariance under positive affine transformations. Practice identifying reference outcomes and distinguishing a rescaling from a substantive change in the utility function.

Sources

Russell's AI teaching slides, utility scales and normalized utilities: https://aima.eecs.berkeley.edu/~russell/classes/cs188/f05/slides/chapter16-short-6pp.pdf

Poole and Mackworth, Preferences and Utility, reference lotteries and utility scaling: https://www.artint.info/3e/html/ArtInt3e.Ch12.S1.html

The supplied glossary interface did not expose its entry; primary author teaching material supplies the expected-utility scope. Numerical comparisons and case are original.

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