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The widely used Plackett-Luce ranking model assumes that individuals rankitems by making repeated choices from a universe of items. But in many casesthe universe is too big for people to plausibly consider all options. In thechoice literature, this issue has been addressed by supposing that individualsfirst sample a small consideration set and then choose among the considereditems. However, inferring unobserved consideration sets (or item considerationprobabilities) in this "consider then choose" setting poses significantchallenges, because even simple models of consideration with strongindependence assumptions are not identifiable, even if item utilities areknown. We apply the consider-then-choose framework to top-$k$ rankings, wherewe assume rankings are constructed according to a Plackett-Luce model aftersampling a consideration set. While item consideration probabilities remainnon-identified in this setting, we prove that we can infer bounds on therelative values of consideration probabilities. Additionally, given a conditionon the expected consideration set size and known item utilities, we deriveabsolute upper and lower bounds on item consideration probabilities. We alsoprovide algorithms 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.

When the Universe is Too Big: Bounding Consideration Probabilities for Plackett-Luce Rankings