When choosing means picking a collection, not a winner

Choosing does not always mean picking one winner. A supermarket stocks several products, a voter may approve multiple candidates, and a platform displays a collection of recommendations. The outcome is a set of options, not necessarily a single favorite.

In the arXiv preprint Random Collection, University of Maryland economist Tri Phu Vu studies how established mathematical models describe these situations. The contribution is theoretical: rather than collecting new behavioral data, the paper proves which observable restrictions correspond to different model specifications.

The central question is practical for anyone using these models: when an equation assigns probabilities to collections, what assumptions about behavior come with it?

Two rules for comparing choices

The framework starts with a finite universe containing at least three options. A menu is the set currently available. For each menu, the model assigns a probability to every collection that might be selected. These probabilities can describe repeated choices by one decision maker or the proportions of people choosing different collections.

Most of the analysis requires the selected collection to be nonempty. An appendix separately examines models that allow choosing nothing—a distinction that changes some of the mathematical restrictions.

The first organizing rule is Independence of Irrelevant Sets. Consider two collections that remain available when a menu changes. Wherever their probabilities are positive, the rule says their relative odds must stay the same. Adding another option can change their individual probabilities, but cannot change how likely one is relative to the other.

The second rule, Relative Additivity, concerns what happens when an option is removed. Suppose that option is a pear. A collection of apples alone and a collection containing those same apples plus the pear both become the same apples-only collection once the pear disappears.

The rule compares the apples-only collection's probability on the smaller menu with the combined probability of those two collections on the larger menu. When the relevant probabilities are positive, this adjustment must have the same proportional size for every nonempty collection on the smaller menu. It is a restriction on probability distributions, not a claim about how an individual physically revises a basket.

Different stories, shared predictions

These rules connect models that initially look quite different.

In the collection Logit model, every nonempty collection has a fixed positive weight. Its selection probability is its weight divided by the total weight of all feasible collections. Vu proves that, under full support—meaning every nonempty feasible collection has positive probability—this model is exactly characterized by Independence of Irrelevant Sets.

Random Categorization tells another story. A decision maker draws a category, selects all its available members, and redraws if none are available. Categories may overlap. Vu characterizes this model through Relative Additivity together with a coverage condition: every available option must appear in some collection selected with positive probability.

Independent Choice starts instead with separate item-level selection probabilities. Items are drawn independently, with empty draws repeated until something is selected. Under full support, Vu proves that this model satisfies both main rules—and that any choice system satisfying both has an Independent Choice representation. It therefore lies precisely at the intersection of Logit and Random Categorization. The paper acknowledges an earlier related intersection result in a different setting.

Other equivalences require careful qualifications. Random Categorization is behaviorally equivalent to a one-time version of Elimination by Aspects when the analyst can freely specify attributes. This does not make it equivalent to the original sequential procedure, which keeps eliminating options until only one remains. Fixed, observed attributes impose additional restrictions.

What the mathematics can—and cannot—establish

The paper also illustrates how relaxing either organizing rule produces richer models. Pairwise interactions between items can preserve fixed relative odds while violating Relative Additivity, or preserve Relative Additivity while changing relative odds. Allowing item-level selection probabilities to depend on the menu can violate both rules and accommodate additional context effects. These are model possibilities, not new experimental findings.

Testing the restrictions requires information about collections across menus. Approval ballots or displayed recommendations may reveal collections directly. Unobserved consideration sets are harder: tracking records, self-reports or final purchase choices provide information only under suitable assumptions.

Finally, identical collection probabilities need not produce identical purchases. If someone subsequently picks one item from the selected collection, that second-stage rule matters too.

The preprint offers a map of assumptions, not evidence that any one model describes people universally. Its value is making explicit what a chosen formula permits, what it rules out, and which observations could contradict it.