Ordinal and Cardinal Utility

Why the Meaning of Utility Numbers Matters

A utility function assigns numbers to alternatives, but the numbers do not automatically measure how much satisfaction a person receives. In many consumer-choice problems, utility is a convenient way to record which bundles a consumer prefers. The order of the numbers matters; their levels and gaps do not. A stronger interpretation is possible when utility differences have meaning under a specified model, but that interpretation requires additional assumptions.

This distinction affects what can be concluded from a calculation. A utility ranking can identify the most-preferred affordable bundle, while a numerical difference between two ordinal utility values does not show how much stronger one preference is. Comparisons of utility changes across people require still more care.

Definition: An ordinal utility representation assigns a number $U(x)$ to each alternative $x$ so that $x$ is at least as preferred as $y$ exactly when $U(x)\geq U(y)$. The numbers encode the ranking of alternatives; their differences and ratios have no required interpretation.

Definition: A cardinal utility representation gives utility differences a specified, meaningful interpretation within a model. For example, in an expected utility model, a person's utility numbers represent preference over lotteries, and expected utility is used to rank those lotteries. A list of numbers is not cardinal merely because subtraction or division is possible.

For ordinary consumer choice among certain bundles, utility is commonly used ordinally. If bundle $x$ is preferred to bundle $y$, the utility assigned to $x$ is higher. The difference $U(x)-U(y)$, however, is not automatically a measure of the intensity of that preference. Nor does $U(x)=20$ mean that the consumer has twice as much satisfaction as at $U(y)=10$.

What Ordinal Information Can Establish

Suppose a consumer can choose only from a feasible set $B$, such as the bundles affordable at given prices and income. A utility representation identifies a most-preferred feasible bundle by finding a bundle with the highest utility in $B$. This conclusion depends on the ordering of feasible bundles, not on the sizes of the utility gaps.

x^* \text{ is most preferred in } B \quad\Longleftrightarrow\quad U(x^*)\geq U(x) \text{ for every } x\in B.

The justification follows directly from the definition of a representation. If $U(x^*)\geq U(x)$ for every feasible $x$, then the represented preference ranking places $x^*$ at least as high as every other feasible bundle. The reverse also holds: if $x^*$ is at least as preferred as every feasible bundle, the representation assigns it utility at least as high as theirs. No utility difference is used in either direction.

List the feasible alternatives.
Only alternatives in the feasible set can be chosen.

Compare their represented rankings.
Find the alternative with the highest ordinal utility value.

Report the choice, not a strength of preference.
The maximizing alternative is identified, but gaps between its utility and other values do not measure how much more it is preferred.

Thus ordinal information can support statements about ranking, indifference, and choice from a given feasible set. It does not, on its own, support statements about the magnitude of satisfaction, the size of a welfare gain, or how much one person benefits relative to another.

Worked Example: Choosing From Affordable Bundles

A consumer has income of $14. Good $x$ costs $3 per unit and good $y$ costs $2 per unit. Three candidate bundles have the following ordinal utility values: $A=(2,3)$, $U(A)=8$; $B=(4,1)$, $U(B)=7$; and $C=(1,4)$, $U(C)=9$. Find the most-preferred affordable candidate and explain what the utility gaps mean.

Method. Check each bundle's cost, then compare utility only among affordable bundles.

For $A$, the cost is $3(2)+2(3)=6+6=\$12$, so $A$ is affordable. For $B$, the cost is $3(4)+2(1)=12+2=\$14$, so $B$ is affordable. For $C$, the cost is $3(1)+2(4)=3+8=\$11$, so $C$ is affordable. Each candidate is within the $14 budget.

The utility ranking is $U(C)=9>U(A)=8>U(B)=7$. Therefore, among these candidates, $C$ is most preferred. The numerical gaps are $9-8=1$ between $C$ and $A$, and $8-7=1$ between $A$ and $B$. Because these are ordinal values, equal gaps do not establish equal preference intensity. The result is the ranking $C\succ A\succ B$, not a measurement of how much better $C$ is.

Check. The costs are $12, $14, and $11, all no greater than $14, and $9$ is the highest utility value. The choice among the listed bundles is $C$.

When Differences Have Meaning

A cardinal interpretation requires a model that explains what utility differences mean. One important case is expected utility for choices among lotteries. A lottery assigns probabilities to possible outcomes. Under the expected utility model, the decision-maker ranks each lottery by the probability-weighted average of the utilities of its outcomes.

The formula applies when the stated utility function is a valid expected utility representation for the decision-maker's preferences over lotteries, and the probabilities describe the actual lottery. An ordinal ranking of certain outcomes alone is not enough to calculate a meaningful expected utility. The numerical utility differences must carry meaning within the expected utility model.

EU(L)=\sum_{j=1}^{n} p_j u(x_j), \qquad \sum_{j=1}^{n}p_j=1.

Here $L$ is a lottery with outcomes $x_j$, probabilities $p_j$, and utility $u(x_j)$ for each outcome. Expected utility is not the same as expected dollar value: it averages utility values, not outcome amounts. The model ranks lotteries by expected utility; it does not say that a utility value is a dollar amount.

Worked Example: Comparing a Sure Payment With a Lottery

A decision-maker faces a sure payment of $100 or a lottery paying $0 with probability $0.5$ and $200 with probability $0.5$. Assume the decision-maker's expected utility representation assigns $u(\$0)=0$, $u(\$100)=4$, and $u(\$200)=10$. Which option is preferred under this model?

Method. Calculate the utility of the sure payment and the probability-weighted utility of the lottery. The stated values are assumed to be cardinal for this expected utility comparison.

For the sure payment, expected utility is simply $u(\$100)=4$.

For the lottery, the probability of each listed outcome is $0.5$, and the probabilities sum to $0.5+0.5=1$. Therefore, $EU(L)=0.5(0)+0.5(10)=0+5=5$.

Since $5>4$, the decision-maker prefers the lottery under the stated expected utility model. Its expected dollar payment is $0.5(\$0)+0.5(\$200)=\$100$, the same as the sure payment, but that dollar calculation is not the comparison used by the model.

Common mistake avoided. Treating the lottery as equivalent to the sure payment because both have an expected dollar value of $100 ignores the specified utility function. The model's comparison is expected utility, $5$ versus $4$. This conclusion depends on the given cardinal expected utility representation; an ordinal ranking of the three sure payment amounts would not, by itself, determine the choice between the lottery and the sure payment.

Check. The lottery's expected utility is $5$, the sure payment's utility is $4$, and the model therefore selects the lottery.

This example illustrates why the distinction matters. For certain outcomes, a ranking tells us which amount the person prefers. A choice between lotteries also depends on how the decision-maker's preferences treat uncertainty and the probabilities of outcomes. Expected utility uses utility differences to represent that additional information.

Utility Differences Within and Across People

Even when a model gives utility differences a meaning for one person, it does not automatically make that person's utility scale comparable with another person's scale. The scales may have different units or origins. To compare utility gains across people, an analyst needs an explicit basis for interpersonal comparability, such as a common scale and assumptions that justify using it.

Worked Example: Interpreting Utility Gains for Two Households

Suppose a study uses a cardinal utility scale for each household separately. Household A's utility rises from 12 to 18 after a policy change; household B's rises from 40 to 45. Find each within-household change and assess whether these values establish which household gained more.

Method. Subtract each household's before value from its after value, then check whether the scales are stated to be comparable.

For household A, the change is $18-12=6$ utility units. For household B, the change is $45-40=5$ utility units. On each household's own scale, the utility level increased.

The figures do not establish that A gained more than B. The problem specifies separate scales, not a shared unit for comparing utility changes across households. For example, if B's scale were expressed in half-sized units, the same before-and-after difference would be $0.5(45-40)=2.5$ units. The individual ranking and direction of B's change would remain, but the numerical size of the change would differ with the unit.

Check. The within-household differences are 6 and 5 on their respective scales. Without a common comparison basis, neither difference can be interpreted as larger in welfare terms across households.

Common Mistake: Treating Ordinal Gaps as Measurements

A frequent error is to subtract ordinal utility values and then interpret the result as a measured benefit. For example, if $U(A)=12$ and $U(B)=8$, the calculation $12-8=4$ is arithmetically correct. The claim that $A$ gives “four units more satisfaction” is not justified by ordinal information alone.

The correction is to state only what the representation supports: $A$ is ranked above $B$. A difference can be interpreted only if the problem specifies a cardinal model that gives the difference meaning. Similarly, saying that $U(A)=12$ represents twice the utility of $U(B)=6$ is not warranted by an ordinal scale.

Valid from ordinal utility: $A$ is preferred to $B$ if $U(A)>U(B)$.
Not valid from ordinal utility alone: the utility gap measures the intensity of preference.
Not valid without a common scale: one person experiences a larger welfare gain than another because that person's utility change is numerically higher.
Valid under the stated expected utility model: compare lotteries by their probability-weighted utilities when the utility numbers are specified for that model.

Distinguishing these interpretations is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. In a calculation, identify whether the data provide only a ranking or a cardinal scale before interpreting utility values or differences.

Key takeaway: Ordinal utility is enough to rank alternatives and identify a preferred feasible choice. Utility differences, utility ratios, and comparisons of welfare gains need additional cardinal and comparability assumptions; expected utility is one model in which cardinal utility differences have a defined role.

Check Your Understanding

For each problem, distinguish what follows from the stated ranking from what would require a cardinal interpretation.

A consumer can afford bundles $D$, $E$, and $F$. Their ordinal utility values are 6, 11, and 9, respectively. Identify the preferred bundle and say whether the difference between the top two values measures preference intensity. (Answer: $E$ is preferred; the gap of 2 does not measure preference intensity.)
A decision-maker has ordinal utility values $U(G)=3$ and $U(H)=9$. What ranking follows, and can one conclude that $H$ provides three times as much satisfaction? (Answer: $H$ ranks above $G$; no three-to-one satisfaction conclusion follows.)
Under an expected utility model, a sure outcome has utility 6. A lottery gives utility 2 with probability $0.25$ and utility 10 with probability $0.75$. Calculate expected utility and state the preferred option under the model. (Answer: $0.25(2)+0.75(10)=0.5+7.5=8$; prefer the lottery.)
Two households have separate cardinal scales. Household J's utility changes from 5 to 9, while household K's changes from 20 to 27. Calculate each change and determine whether the figures alone show which household gained more. (Answer: J's change is 4 and K's is 7 scale units; the figures do not establish which gained more across households.)
A lottery pays $0 or $120, each with probability $0.5$. A sure payment is $60. Under a specified expected utility representation, $u(\$0)=0$, $u(\$60)=3$, and $u(\$120)=8$. Compare the options, and state why comparing only expected dollar payments would not settle the model's utility comparison. (Answer: Lottery expected utility is $0.5(0)+0.5(8)=4$, versus 3 for the sure payment, so the lottery is preferred; both have expected payment $60, but expected utility is the specified comparison.)

What You'll Learn