Partially Identified Rankings from Pairwise Interactions

TL;DR

This paper investigates how to infer object rankings from pairwise interaction data, especially when observations are incomplete, highlighting differences between parametric and nonparametric models and proposing methods for testing ranking consistency.

Contribution

It characterizes the identified set of rankings in nonparametric models with incomplete data and introduces likelihood-based tests for ranking validity.

Findings

Rankings are point identified in parametric models with sparse data.

Nonparametric models require dense interaction graphs for identification.

Proposed tests can validate if a ranking fits within the identified set.

Abstract

This paper considers the problem of ranking objects based on their latent merits using data from pairwise interactions. We allow for incomplete observation of these interactions and study what can be inferred about rankings in such settings. First, we show that identification of the ranking depends on a trade-off between the tournament graph and the interaction function: in parametric models, such as the Bradley-Terry-Luce, rankings are point identified even with sparse graphs, whereas nonparametric models require dense graphs. Second, moving beyond point identification, we characterize the identified set in the nonparametric model under any tournament structure and represent it through moment inequalities. Finally, we propose a likelihood-based statistic to test whether a ranking belongs to the identified set. We study two testing procedures: one is finite-sample valid but computationally intensive; the other is easy to implement and valid asymptotically. We illustrate our results using Brazilian employer-employee data to study how workers rank firms when moving across jobs.