Bounding Consideration Probabilities in Consider-Then-Choose Ranking
  Models
Ben Aoki-Sherwood Catherine Bregou David Liben-Nowell Kiran Tomlinson Thomas Zeng
A common theory of choice posits that individuals make choices in a two-step
process, first selecting some subset of the alternatives to consider before
making a selection from the resulting consideration set. However, inferring
unobserved consideration sets (or item consideration probabilities) in this
"consider then choose" setting poses significant challenges, because even
simple models of consideration with strong independence assumptions are not
identifiable, even if item utilities are known. We consider a natural extension
of consider-then-choose models to a top-$k$ ranking setting, where we assume
rankings are constructed according to a Plackett-Luce model after sampling a
consideration set. While item consideration probabilities remain non-identified
in this setting, we prove that knowledge of item utilities allows us to infer
bounds on the relative sizes of consideration probabilities. Additionally,
given a condition on the expected consideration set size, we derive absolute
upper and lower bounds on item consideration probabilities. We also provide
algorithms to tighten those bounds on consideration probabilities by
propagating inferred constraints. Thus, we show that we can learn useful
information about consideration probabilities despite not being able to
identify them precisely. We demonstrate our methods on a ranking dataset from a
psychology experiment with two different ranking tasks (one with fixed
consideration sets and one with unknown consideration sets). This combination
of data allows us to estimate utilities and then learn about unknown
consideration probabilities using our bounds.