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

arXiv: https://arxiv.org/abs/2401.11016 · HTML: https://mlsanity.com/paper/2401.11016/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 59 references that are in the map.
- First posted on arXiv 19 Jan 2024.

## Builds on (most cited first, 7 of 7)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Learning Mixtures of Plackett-Luce Models](https://mlsanity.com/paper/1603.07323/) (2016)
- [From PAC to Instance-Optimal Sample Complexity in the Plackett-Luce Model](https://mlsanity.com/paper/1903.00558/) (2019)
- [Learning Mixtures of Plackett-Luce Models from Structured Partial Orders](https://mlsanity.com/paper/1910.11721/) (2019)
- [Learning Rich Rankings](https://mlsanity.com/paper/2312.15081/) (2023)
- [Efficient and Accurate Learning of Mixtures of Plackett-Luce Models](https://mlsanity.com/paper/2302.05343/) (2023)
- [A Random Attention Model](https://mlsanity.com/paper/1712.03448/) (2017)

## Built on by (most cited first, 1 of 1)

- [Covariate-dependent Joint Modeling of Multivariate Ordinal Preferences and Its Connections with Comparison Models](https://mlsanity.com/paper/2610.09070/) (2026)

## Abstract (arXiv)

The widely used Plackett-Luce ranking model assumes that individuals rank items by making repeated choices from a universe of items. But in many cases the universe is too big for people to plausibly consider all options. In the choice literature, this issue has been addressed by supposing that individuals first sample a small consideration set and then choose among the considered items. 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 apply the consider-then-choose framework to top-k rankings, 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 we can infer bounds on the relative values of consideration probabilities. Additionally, given a condition on the expected consideration set size and known item utilities, 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.