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A common theory of choice posits that individuals make choices in a two-stepprocess, first selecting some subset of the alternatives to consider beforemaking a selection from the resulting consideration set. However, inferringunobserved consideration sets (or item consideration probabilities) in this"consider then choose" setting poses significant challenges, because evensimple models of consideration with strong independence assumptions are notidentifiable, even if item utilities are known. We consider a natural extensionof consider-then-choose models to a top-$k$ ranking setting, where we assumerankings are constructed according to a Plackett-Luce model after sampling aconsideration set. While item consideration probabilities remain non-identifiedin this setting, we prove that knowledge of item utilities allows us to inferbounds on the relative sizes of consideration probabilities. Additionally,given a condition on the expected consideration set size, we derive absoluteupper and lower bounds on item consideration probabilities. We also providealgorithms to tighten those bounds on consideration probabilities bypropagating inferred constraints. Thus, we show that we can learn usefulinformation about consideration probabilities despite not being able toidentify them precisely. We demonstrate our methods on a ranking dataset from apsychology experiment with two different ranking tasks (one with fixedconsideration sets and one with unknown consideration sets). This combinationof data allows us to estimate utilities and then learn about unknownconsideration probabilities using our bounds.

Bounding Consideration Probabilities in Consider-Then-Choose Ranking Models