Utility Functions and Preference Rankings

Why Represent Preferences With Numbers?

A consumer may compare bundles of goods without attaching a numerical score to each one. For example, a person might prefer bundle A to bundle B, regard bundles B and C as equally desirable, and rank both above bundle D. A utility function is a way to assign numbers to bundles so that comparing those numbers reproduces the consumer’s stated ranking.

The key test is not whether the numbers seem large, small, or realistic. It is whether their ordering matches every preference comparison, including any ties. A candidate assignment that gets most comparisons right but reverses one pair does not represent the stated ranking.

Definition: A bundle is a specified combination of goods available for a consumer to compare. The relation $x \succeq y$ means bundle $x$ is weakly preferred to bundle $y$: the consumer prefers $x$, or regards the two bundles as equally good. The relation $x \succ y$ means $x$ is strictly preferred to $y$. The relation $x \sim y$ means the consumer is indifferent between them; equivalently, $x \succeq y$ and $y \succeq x$.

The Test for a Utility Representation

A utility function $U$ assigns a real number to each bundle in the set being considered. It represents the consumer’s preferences if, for every pair of bundles $x$ and $y$, the number assigned to $x$ is at least the number assigned to $y$ exactly when $x$ is weakly preferred to $y$.

Definition: A function $U$ is a utility representation of the preference relation $\succeq$ when, for every pair $x,y$, $$ x \succeq y \quad\Longleftrightarrow\quad U(x) \geq U(y). $$ Thus $x \succ y$ requires $U(x)>U(y)$, and $x\sim y$ requires $U(x)=U(y)$.

The phrase “if and only if” matters. If $x\succ y$, then $U(x)>U(y)$ must hold; but it must also be true that a higher utility number does not imply a preference contrary to the stated ranking. If the consumer is indifferent between two bundles, they must receive the same number. Merely assigning numbers that increase for some preferred bundles is not enough.

For a finite set of bundles with a complete, consistent ranking, the practical procedure is to list the ranking from most preferred to least preferred, group indifferent bundles together, and check that utility levels descend strictly from one group to the next. Bundles within a tie group must have identical utility. The actual numerical gaps between groups do not determine whether the assignment passes this test.

x \succ y \Longleftrightarrow U(x)>U(y) \qquad\text{and}\qquad x \sim y \Longleftrightarrow U(x)=U(y)

Worked Examples: Checking Rankings and Ties

Worked Example: Testing a Complete Ranking

A consumer ranks four hypothetical meal bundles as $A\succ C\succ B\sim D$. A proposed assignment is $U(A)=18$, $U(B)=4$, $U(C)=11$, and $U(D)=4$. Does the assignment represent the stated ranking?

Method. Compare the utility values in the order of the stated preferences, and check that the indifferent bundles receive equal values.

The ranking requires $A$ above $C$, $C$ above both $B$ and $D$, and $B$ tied with $D$. The proposed values give $18>11$, so $U(A)>U(C)$. They also give $11>4$, so $U(C)>U(B)$ and $U(C)>U(D)$. Finally, $U(B)=4=U(D)$, as required for indifference.

These checks account for all six pairwise comparisons among four bundles: the comparisons of $A$ with $C,B,D$; of $C$ with $B,D$; and of $B$ with $D$. The resulting ordering is exactly $A\succ C\succ B\sim D$. Therefore, the assignment does represent the stated ranking.

Trap avoided. Do not conclude that the assignment works just because the most preferred bundle has the highest number. The tie between $B$ and $D$ must also be represented correctly.

Worked Example: Detecting a Broken Tie

For five hypothetical travel bundles, the stated ranking is $P\sim Q\succ R\succ S\sim T$. A proposed assignment is $U(P)=12$, $U(Q)=12$, $U(R)=7$, $U(S)=3$, and $U(T)=2$. Does it represent the ranking?

Method. First compare utility values within each indifference group. Then compare the levels of the groups.

For the top group, $U(P)=12=U(Q)$, so the tie between $P$ and $Q$ is represented. Both exceed $U(R)=7$, giving $P\succ R$ and $Q\succ R$, as required. Also, $7>3$ and $7>2$, so $R$ is ranked above both $S$ and $T$.

However, the stated ranking requires $S\sim T$, which means their utility values must be equal. The assignment gives $U(S)=3$ and $U(T)=2$, so it instead represents $S\succ T$. Therefore, the assignment does not represent the full ranking.

A corrected assignment could keep the first three values and set $U(T)=3$. Then $P$ and $Q$ tie at 12, both exceed $R$ at 7, and $S$ and $T$ tie at 3 below $R$. The corrected assignment represents $P\sim Q\succ R\succ S\sim T$.

Trap avoided. Putting two indifferent bundles in the same general rank range is insufficient. A utility representation requires equality of their assigned values.

Why the Comparison Rule Works

The representation condition translates each preference comparison directly into a numerical comparison. This gives a short way to verify why checking both strict rankings and ties is necessary. Suppose the rule holds for every pair of bundles.

Translate weak preference.
By definition of representation, $x\succeq y$ holds exactly when $U(x)\geq U(y)$.

Translate strict preference.
Strict preference means $x\succeq y$ but not $y\succeq x$. The first comparison gives $U(x)\geq U(y)$; the failure of the reverse comparison gives $U(y)<U(x)$. Together these imply $U(x)>U(y)$.

Translate indifference.
Indifference means both $x\succeq y$ and $y\succeq x$. The representation rule gives both $U(x)\geq U(y)$ and $U(y)\geq U(x)$, which together imply $U(x)=U(y)$.

The reverse checks follow from the same equivalence: if $U(x)>U(y)$, then $U(x)\geq U(y)$ but not $U(y)\geq U(x)$, so $x\succ y$. If $U(x)=U(y)$, both weak comparisons hold, so $x\sim y$. Thus preserving strict rankings and ties is not an extra convention; it follows from the definition of representation.

Worked Example: A Preference Cycle Cannot Be Represented

A set of three hypothetical subscription bundles is described by the strict preferences $A\succ B$, $B\succ C$, and $C\succ A$. A proposed assignment is $U(A)=4$, $U(B)=3$, and $U(C)=5$. Can any utility assignment represent all three preferences?

Method. Convert each strict preference into a strict inequality and test whether the inequalities can all hold at once.

The first preference requires $U(A)>U(B)$, and the second requires $U(B)>U(C)$. Combining them gives $U(A)>U(C)$. But the third preference, $C\succ A$, requires $U(C)>U(A)$. A pair of real numbers cannot satisfy both $U(A)>U(C)$ and $U(C)>U(A)$. Therefore, no utility function can represent all three stated comparisons.

The proposed values illustrate the conflict: $4>3$ satisfies $A\succ B$, and $3<5$ violates $B\succ C$; moreover, $5>4$ satisfies $C\succ A$. The assignment fails, and changing its numbers cannot fix the underlying cycle. The problem is the set of stated preferences, not the particular candidate values.

Trap avoided. Do not try to repair an inconsistent cycle by choosing more widely separated numbers. Strict inequalities around a cycle still contradict one another, regardless of their numerical size.

A Reliable Checking Procedure

For a proposed assignment on a finite set, the following procedure is efficient and reduces the risk of missing a tie or a reversed comparison.

Write the stated ranking clearly. Separate strict rankings from indifference groups. For example, $A\succ B\sim C\succ D$ has three levels: $A$, then the tied pair $B,C$, then $D$.
Compare utility within each group. Bundles in the same indifference group must have equal utility values.
Compare utility across groups. Every bundle in a higher-ranked group must have a strictly greater utility value than every bundle in a lower-ranked group.
Check all stated comparisons. If the description gives separate pairwise statements rather than one complete ranking, translate each statement into its required equality or inequality. Do not infer that an unstated comparison is known.
Conclude explicitly. Say whether the assignment represents the ranking and identify any specific violated comparison. If the preference statements themselves imply contradictory inequalities, explain why no assignment can work.

A useful shortcut is to sort bundles by their proposed utility values and then compare the resulting order, including ties, with the stated order. This is safe only if all relevant bundles are included and equal values are checked as ties. If two bundles receive different values, the utility function ranks one strictly above the other, even if the difference is small.

Common Mistakes and Why the Test Matters

Checking only the highest-ranked bundle. A candidate can give the top bundle the largest number and still reverse two lower bundles or break an indifference.
Treating a tie as “almost equal.” If the preference statement says $x\sim y$, representation requires exact equality $U(x)=U(y)$, not merely close numerical values.
Ignoring the reverse implication. If a candidate assigns a larger number to $x$ than to $y$, it represents $x\succ y$. It cannot represent $y\succ x$ or $x\sim y$.
Confusing a numerical score with evidence about preferences. A list of utility values is a candidate representation. It must be tested against the stated preferences; the numbers alone do not establish what a consumer prefers.
Trying to solve a logical contradiction numerically. If the stated strict comparisons form a cycle, the resulting inequalities are impossible for real numbers. Changing the utility scale cannot remove the contradiction.

The representation test is useful because it turns verbal rankings into a precise consistency check. For the finite comparisons commonly used in applications and exam-style problems, a correct assignment must preserve every strict preference and every indifference. This tutorial is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Check Your Understanding

For each problem, check both the ordering and any stated ties before deciding whether the utility assignment represents the ranking.

A consumer ranks $A\succ B\sim C\succ D$. A candidate assignment gives $U(A)=15$, $U(B)=8$, $U(C)=8$, and $U(D)=2$. Does it represent the ranking? (Answer: Yes; $15>8=8>2$.)
For bundles $K,L,M$, the stated ranking is $K\sim L\succ M$. The proposed values are $U(K)=9$, $U(L)=8$, and $U(M)=3$. Identify the violated condition. (Answer: No; $K\sim L$ requires equal utility, but $9\ne8$.)
A stated ranking is $R\succ S\succ T$, and a proposed assignment gives $U(R)=6$, $U(S)=10$, and $U(T)=2$. Does it represent the ranking? (Answer: No; $R\succ S$ requires $U(R)>U(S)$, but $6<10$.)
Suppose the stated comparisons are $G\succ H$, $H\succ J$, and $J\succ G$. Can a real-valued utility function represent them? Show the implied conflict. (Answer: No; they require $U(G)>U(H)>U(J)>U(G)$, which is impossible.)
Five bundles have the stated ranking $P\succ Q\sim R\succ S\sim T$. A candidate assigns $U(P)=20$, $U(Q)=11$, $U(R)=11$, $U(S)=4$, and $U(T)=4$. State whether it works and give one valid alternative assignment. (Answer: Yes; it represents the ranking. For example, $U(P)=3$, $U(Q)=2$, $U(R)=2$, $U(S)=0$, and $U(T)=0$ also works.)

What You'll Learn