Completeness as a Ranking Axiom

Why Completeness Matters

A consumer choice model needs a way to describe how bundles compare. If a model is to represent a consumer choosing between any two available bundles, its preference relation must say whether one bundle is at least as good as the other—or the other is at least as good as the first. A relation that leaves some pairs unranked may still describe some comparisons, but it does not provide a ranking for every possible choice.

Completeness is this pairwise coverage requirement. It does not say that the consumer likes more of every good, or that comparisons across different pairs fit together consistently. It asks only whether each pair can be compared in at least one direction.

Definition: Let \(X\) be a set of consumption bundles, and let \(x \succeq y\) mean that bundle \(x\) is weakly preferred to bundle \(y\). The preference relation \(\succeq\) is complete on \(X\) if, for every pair \(x,y \in X\), at least one of the following holds: \(x \succeq y\) or \(y \succeq x\). For distinct bundles, a pair is incomplete only when neither direction holds.

The weak-preference symbol matters: completeness requires a weak ranking in at least one direction, not necessarily a strict preference. If both directions hold, the consumer is indifferent between the bundles. If exactly one direction holds, one bundle is weakly preferred while the reverse weak preference does not hold; under the standard definitions, that is strict preference in that direction.

A relation is a formal description of comparisons, not a statement about what an analyst happens to know. If the model specifies no direction for a pair, that pair is unranked within the model. Lack of survey evidence about a consumer’s view is a different issue: it is uncertainty for the analyst, not by itself a failure of completeness in a specified preference relation.

How to Test a Preference Relation

For a finite set of bundles, list every distinct pair once and check whether at least one weak-preference direction is included in the relation. A pair does not pass just because the bundles differ, or because a decision-maker could be asked to choose between them. The stated relation itself must supply at least one direction.

Identify the domain.
Write down the set \(X\) of bundles being considered. Completeness is assessed on that set.

List distinct pairs.
For each pair \(x,y\), check both possible directions: \(x \succeq y\) and \(y \succeq x\).

Look for a missing pair.
If at least one distinct pair has neither direction, the relation is incomplete on \(X\). If every pair has at least one direction, it is complete on \(X\).

Worked Example: Finding an Unranked Pair in a Finite Relation

A planner considers three hypothetical bundles of household supplies: \(A\), \(B\), and \(C\). The stated preference relation contains the comparisons \(A \succeq A\), \(B \succeq B\), \(C \succeq C\), \(A \succeq B\), \(B \succeq A\), and \(A \succeq C\), and no other comparisons. Is the relation complete on these three bundles?

Method. List the distinct unordered pairs: \((A,B)\), \((A,C)\), and \((B,C)\). Completeness requires at least one direction for each pair.

For \((A,B)\), both \(A \succeq B\) and \(B \succeq A\) are included. This pair is ranked in both directions, so \(A\) and \(B\) are indifferent under the relation.

For \((A,C)\), \(A \succeq C\) is included. At least one direction is present, so this pair meets the completeness requirement even though \(C \succeq A\) is not listed.

For \((B,C)\), neither \(B \succeq C\) nor \(C \succeq B\) is included. This is an unranked pair. Therefore, the relation is not complete on \(\{A,B,C\}\).

Common mistake avoided: Treating the two directions for \((A,B)\) as necessary for completeness. Only one direction is required. The failure is the missing comparison for \((B,C)\).

Comparing Bundles by Their Components

A comparison rule may assess bundles using quantities of more than one good. One possible rule ranks \(x\) at least as high as \(y\) only when \(x\) contains at least as much of every good as \(y\). This coordinate-by-coordinate rule can leave trade-off pairs unranked: one bundle may contain more of one good but less of another.

Definition: For two-good bundles \(x=(x_1,x_2)\) and \(y=(y_1,y_2)\), the coordinate-by-coordinate weak comparison rule declares \(x \succeq y\) when \(x_1 \geq y_1\) and \(x_2 \geq y_2\). A pair fails the completeness test under this rule if neither bundle has at least as much of both goods as the other.

Worked Example: Detecting a Trade-Off Pair

A model compares hypothetical weekly bundles of tea packets and transit tickets using the coordinate-by-coordinate rule. The available bundles are \(A=(6,1)\), \(B=(4,3)\), and \(C=(5,0)\), where the first coordinate is tea packets and the second is tickets. Determine whether the rule is complete on these three bundles.

Method. Apply the rule to each distinct pair. For \(x \succeq y\), both coordinates of \(x\) must be at least the corresponding coordinates of \(y\).

Compare \(A=(6,1)\) and \(B=(4,3)\). For \(A \succeq B\), the first-coordinate test passes because \(6 \geq 4\), but the second fails because \(1 \not\geq 3\). Thus \(A \not\succeq B\). For \(B \succeq A\), the first-coordinate test fails because \(4 \not\geq 6\). Thus \(B \not\succeq A\). The pair is unranked.

Compare \(A=(6,1)\) and \(C=(5,0)\). Since \(6 \geq 5\) and \(1 \geq 0\), \(A \succeq C\). The pair passes, regardless of whether the reverse direction holds.

Compare \(B=(4,3)\) and \(C=(5,0)\). For \(B \succeq C\), the first coordinate fails because \(4 \not\geq 5\). For \(C \succeq B\), the second coordinate fails because \(0 \not\geq 3\). Neither direction holds.

There are two unranked pairs, \((A,B)\) and \((B,C)\). The relation is therefore incomplete on the stated set.

Common mistake avoided: Concluding that a pair is ranked because one bundle has more of one good. The rule requires at least as much of both goods in the same direction.

A Tie-Breaker Can Make a Rule Complete

A comparison rule can handle trade-offs by specifying which attribute takes priority. For example, suppose a model ranks bundles first by battery-hours and uses warranty months only when battery-hours are equal. The priority rule makes a comparison possible even when one bundle has more warranty but fewer battery-hours.

Definition: For bundles \(x=(x_1,x_2)\) and \(y=(y_1,y_2)\), a lexicographic rule with the first coordinate as the priority attribute declares \(x \succeq y\) if \(x_1>y_1\), or if \(x_1=y_1\) and \(x_2 \geq y_2\). If the first coordinates differ, the second coordinate does not decide the comparison. If the first coordinates tie, the second coordinate breaks the tie.

Worked Example: Testing a Priority-Based Ranking

A hypothetical equipment buyer compares three bundles of portable power equipment. Each bundle is described by battery-hours and warranty months. The buyer uses the lexicographic rule above, with battery-hours first: \(A=(8,12)\), \(B=(8,18)\), and \(C=(9,6)\). Check all pairs and determine whether the relation is complete on this set.

Method. Compare battery-hours first. Only when the battery-hour quantities are equal do warranty months determine the direction.

For \(A\) and \(B\), both have 8 battery-hours. Compare warranty months: \(18 \geq 12\), so \(B \succeq A\). The reverse comparison does not hold because \(12 \not\geq 18\).

For \(C\) and \(A\), \(C\) has 9 battery-hours and \(A\) has 8. The first coordinate decides the comparison, so \(C \succeq A\), even though \(C\) has fewer warranty months.

For \(C\) and \(B\), \(C\) has 9 battery-hours and \(B\) has 8. Again the first coordinate decides the comparison, so \(C \succeq B\), despite \(C\)'s shorter warranty.

Every distinct pair has at least one direction. The relation is complete on the three bundles.

Common mistake avoided: Treating a tie-breaker as if both attributes must favor the same bundle. Under this rule, the first coordinate takes priority; the second matters only when the first coordinates tie.

Why the Completeness Condition Works

The condition is a direct translation of “every pair can be compared.” Take any two bundles \(x\) and \(y\). If \(x \succeq y\), the pair has a ranking in the first direction. If that is not the case, completeness requires \(y \succeq x\). Consequently, for each pair, the two possible weak-preference statements cannot both be false.

This also explains why a tie is not a failure. If \(x \succeq y\) and \(y \succeq x\), both possible weak comparisons hold. The pair is ranked in both directions, which represents indifference rather than an unranked comparison. By contrast, if neither statement holds, the relation has no ranking for that pair and completeness fails.

For the lexicographic rule, the same reasoning can be checked from its definition. For any two bundles, either their first coordinates differ or they are equal. If they differ, the larger first coordinate determines a weak-preference direction. If they are equal, the second coordinates either differ, in which case the larger one determines a direction, or are equal, in which case both directions hold. In all cases, at least one direction exists. This establishes completeness for that rule without requiring the bundles to agree on every attribute.

What Completeness Does Not Tell You

Completeness is a condition on comparisons one pair at a time. It does not say which bundle is preferred, how strong a preference is, or why the consumer ranks it that way. Nor does completeness by itself ensure that comparisons among several bundles fit together consistently; that is a separate property of a preference relation.

When evaluating a question, keep the scope precise:

To show a relation is incomplete, give one pair \(x,y\) for which neither \(x \succeq y\) nor \(y \succeq x\) holds.
To establish completeness on a finite set, check every distinct pair, not just a few representative comparisons.
Do not require both directions for every pair; one direction is enough, and both directions represent indifference.
Do not infer completeness just because each bundle has a numerical description. The comparison rule, not the presence of numbers, determines whether pairs are ranked.

Key takeaway: A preference relation is complete exactly when no pair in its domain is left without a weak-preference direction. Completeness permits one-way rankings and indifference, but it rules out incomparability within the stated set.

Check Your Understanding

For each question, use the stated domain and comparison rule; check both directions whenever a pair is at issue.

A relation on \(P,Q,R\) contains \(P \succeq Q\), \(Q \succeq P\), \(Q \succeq R\), and all three self-comparisons, but neither comparison between \(P\) and \(R\). Is it complete? (Answer: No; neither \(P \succeq R\) nor \(R \succeq P\) holds.)
Under a coordinate-by-coordinate rule, two snack bundles are \(M=(3,5)\) and \(N=(2,4)\). State the weak-preference direction or directions and say whether this pair passes the completeness test. (Answer: \(M \succeq N\) only; the pair passes.)
A coordinate-by-coordinate relation is applied to \(U=(7,2)\), \(V=(5,4)\), and \(W=(6,1)\). Is it complete on these bundles? Identify one pair that determines your answer. (Answer: No; \(U\) and \(V\) are unranked because each has less of one good than the other.)
A buyer uses a lexicographic rule that prioritizes delivery days, with fewer days ranked higher; if delivery times tie, lower fees are ranked higher. Compare \(J=(2,40)\) and \(K=(3,20)\), where the first coordinate is days and the second is dollars. Which bundle is weakly preferred? (Answer: \(J \succeq K\), because 2 delivery days is fewer than 3, so the priority attribute decides.)
A relation on \(D,E,F\) has \(D \succeq E\), \(E \succeq D\), \(E \succeq F\), and \(F \succeq E\), along with all self-comparisons; no comparison between \(D\) and \(F\) is specified. Is the relation complete? (Answer: No; the pair \(D,F\) has neither direction.)

What You'll Learn