PlackettLuce

The Plackett-Luce model is a probabilistic framework for modeling ranked data. It describes how a set of items is ordered by a decision-maker, using positive parameters that reflect the relative desirability or “utility” of each item. The model is named after Robyn Plackett and R. Duncan Luce, and it provides a convenient way to express the probability of a complete ranking.

For a complete ranking π of N items, with parameters θ_i > 0 for each item i, the probability under the Plackett-Luce model is

P(π) = ∏_{t=1}^{N} θ_{π_t} / ∑_{j∈R_t} θ_j,

where R_t is the set of items not yet selected at stage t. This formulation yields first-choice probabilities proportional to θ_i, and, conditional on earlier choices, second-stage probabilities in a consistent manner. To ensure identifiability, the θ parameters are typically constrained, for example by normalizing so that sum_i θ_i = 1 or by fixing one θ as a reference.

Estimation and extensions: Parameters are estimated by maximum likelihood from observed rankings, often via iterative algorithms such as minorization-maximization or expectation-maximization, particularly in mixtures of Plackett-Luce models. The model naturally accommodates partial rankings (top-k lists) by using the corresponding truncated likelihood. Extensions address ties, heterogeneity, and dynamics.

Applications and variants: The Plackett-Luce model is a staple in the analysis of preference data and is related to the Bradley-Terry family for rankings. It has been extended to mixtures, Bayesian formulations, and generalized versions to handle ties, time-varying preferences, and partial rankings. It is widely used in marketing research, sports analytics, election studies, and recommender systems where ranked data arise.