By assumption, individual preferences must be transitive so that if A is preferred to B, and B is preferred to C, then A is preferred to C. Suppose that Marsha, Jan, and Cindy individually have transitive preferences over three goods: oranges, apples, and pears. If Marsha, Jan, and Cindy were to vote on whether to name oranges, apples, or pears the “fruit of the month,” Are the preferences for the group necessarily transitive?

Question

By assumption, individual preferences must be transitive so that if A is preferred to B, and B is preferred to C, then A is preferred to C. Suppose that Marsha, Jan, and Cindy individually have transitive preferences over three goods: oranges, apples, and pears. If Marsha, Jan, and Cindy were to vote on whether to name oranges, apples, or pears the “fruit of the month,”  Are the preferences for the group necessarily transitive?

BlackTom AI · Accepted Answer

The question asks whether a group’s collective preferences must be transitive given that each individual (Marsha, Jan, and Cindy) has transitive preferences over three goods (oranges, apples, pears) and the group votes on which fruit is the ‘fruit of the month.’
Option 1: Yes. This would claim that collective preferences under a voting rule preserve transitivity automatically. However, this is not guaranteed. Even if all individuals’ preferences are individually transitive, the aggregation process (e.g., majority voting) can produce cycles where A is preferred to B by a majority, B is preferred to C by a majority, and C is preferred to A by a majority. Such a Condorcet cycle shows intransitivity at the group level.
Option 2: No. This is the correct general conclusion. The classic Condorcet paradox demonstrates that with three or more alternatives and three or more voters, even when each voter has a transitive ranking, the majority-preferred relation can be non-transitive. Different pairwise comparisons can yield a majority that forms a rock-paper-scissors loop rather than a consistent ranking.
To illustrate why the “Yes” claim fails, imagine a scenario where:
- A is preferred to B by two voters, and B to C by two voters, while C is preferred to A by two voters in different pairwise comparisons depending on the voters’ rankings. This creates a cycle in the group’s ranking even though each individual’s ordering is transitive.
Option 1 thus ignores the possibility of such majority cycles; it implicitly assumes the social choice rule preserves transitivity, which is not guaranteed.
Option 2 correctly highlights that transitivity of individuals does not ensure transitivity for the group as a whole under typical aggregation methods like simple majority voting.
In sum, while individual transitivity is a given, the group’s aggregated preference need not be transitive under standard voting rules. The no-ansatz caveat remains: if all individuals exactly agree on a single linear order or if the aggregation rule imposes a transitive social ordering (which some rules attempt, but not guaranteed in general), then transitivity could hold, but that is a special case, not the general rule.

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