Plackett-Luce Model

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What is the Plackett-Luce model?

The Plackett-Luce model, developed by Plackett [1] and Luce [2], can model how people can choose one item out of a set; it can define the most probable rankings and elements in specific places on the ranked list.

Rankings data consists of ordered observations for a set of items and is commonly encountered in various applications such as sports tournaments and consumer studies. For instance, rankings data can include the finishing order of race competitors or the preferences of consumers among competing products.

The Plackett-Luce model defines a probability distribution on permutations of objects, called permutation probability, and assumes the probability of selecting an item i is determined by the value of i compared to the sum of values of all items. The idea is that the model chooses one item of the highest value, then repeats the procedure to select another item, and so on. Luce’s Axoim of Choice governs the choice probabilities of a population of choosers.

Luce’s Axoim of Choice

The Plackett-Luce model is based on Luce’s axiom of choice [2, 3], which states that the choice of one item over another is independent of the set of available items. The model estimates the value of each item in the rankings, with the parameters typically presented in logarithmic scale for inference.

Let’s say we have a set of J unique items [4]:

S = {i₁, i₂, …, i_j}.

According to Luce’s axiom, the probability of choosing an item j from S is given by:

where α_i represents the worth of item i. If we consider the ranking of J items as a sequence of choices — starting with the top-ranked item and then choosing the second-ranked item from the remaining options, and so on — the probability of the ranking i₁ ≻ … ≻ i_j  (note that ≻ is the strict preference relation) is:

Here, A_j represents the set of alternatives {i_j, i_{j + 1}, …, i_J}. from which item i_j is chosen.

This model, known as the Plackett-Luce model, was also derived by Plackett in 1975.

Scale invariance of the Plackett-Luce model

The Plackett-Luce model is scale invariant and translation variant under certain conditions. for example, if the exponential function is used as the transformation function (which is commonly used to ensure positive, non-zero values), after adding the same constant to all the ranking scores, the permutation probability distribution define by the model will not change [5].

References

1. Plackett, Robert L. 1975. “The Analysis of Permutations.” Appl. Statist 24 (2): 193–202. https://doi.org/10.2307/2346567.
2. Luce, R. Duncan. 1959. Individual Choice Behavior: A Theoretical Analysis. New York: Wiley.
3. Luce, R. Duncan. 1977. “The choice axiom after twenty years.” Journal of Mathematical Psychology 15 (3): 215–33. https://doi.org/10.1016/0022-2496(77)90032-3.
4. Turner, H. et al (2023). Introduction to PlackettLuce. Retrieved July 31, 2023 from: https://cran.rstudio.com/web/packages/PlackettLuce/vignettes/Overview.html
5. Liu, T. (2011) Learning to Rank for Information Retrieval. Springer