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Plackett-Luce Models

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Plackett-Luce Models#

Plackett-Luce models are parameterised by a vector of quality for the candidates. The higher the quality of a candidate, the higher the change that they show up high in the rankings.

plackett_luce(num_voters: int, num_candidates: int, alphas: list[float], seed: int = None) → ndarray[source]#

Generates ordinal votes according to Plackett-Luce model.

This model is parameterised by a vector alphas intuitively indicating a quality for each candidate. A vote is generated in m steps (m being the number of candidates). The vote is filled up from most preferred to least preferred candidate. For the initial draw, the probability of selecting a candidate is equal to its quality (normalised). Then, this candidate is removed and the quality are re-normalised.

A collection of num_voters vote is generated independently and identically following the process described above.

For a similar model, see the didi() model.

Parameters:

  • num_voters (int) – Number of Voters.

  • num_candidates (int) – Number of Candidates.

  • alphas (list[float]) – List of model parameters (quality of the candidates).

  • seed (int, default: None) – Seed for numpy random number generator.

Returns:

Ordinal votes.

Return type:

np.ndarray

Examples

from prefsampling.ordinal import plackett_luce

# Sample from a Plackett-Luce model with 2 voters and 3 candidates, the qualities of
# candidates are 0.5, 0.2, and 0.1.
plackett_luce(2, 3, (0.5, 0.2, 0.1))

# For reproducibility, you can set the seed.
plackett_luce(2, 3, (5, 2, 0.7), seed=1002)

# Don't forget to provide a quality for all candidates
try:
    plackett_luce(2, 3, (0.5, 0.2))
except ValueError:
    pass

# And all quality scores need to be strictly positive
try:
    plackett_luce(2, 3, (0.5, 0.2, -0.4))
except ValueError:
    pass
try:
    plackett_luce(2, 3, (0.5, 0.2, 0))
except ValueError:
    pass

Validation

The probability distribution governing the Plackett-Luce model is well documented. Specifically, given n agents and m candidates and a vector of candidates qualities \mathbf{\alpha} = (\alpha_1, \ldots, \alpha_m), the probability of generating a ranking a_{i_1} \succ a_{i_2} \succ \cdots \succ a_{i_m} is equal to:

\frac{\alpha_{i_1}}{1} \times \frac{\alpha_{i_2}}{\sum_{p > 1}\alpha_{i_p}} \times \cdots \times \frac{\alpha_{i_{m-1}}}{\alpha_{i_{m-1}} + \alpha_{i_m}}.

We test that the observed frequencies of rankings aligns with the theoretical probability distribution.

Observed versus theoretical frequencies for a Plackett-Luce model with alpha=[0.62935964 0.23667638 0.03172772 0.96967788 0.83727159] Observed versus theoretical frequencies for a Plackett-Luce model with alpha=[1.0, 0.3, 0.3, 0.3, 0.3]

When all values in alphas are equal, we are supposed to observe a uniform distribution over all rankings.

Observed versus theoretical frequencies for a Plackett-Luce model with alpha=[0.1, 0.1, 0.1, 0.1, 0.1]

References

Individual Choice Behavior: A Theoretical Analysis, R. Duncan Luce, New York: Wiley, 1959.

The Analysis of Permutations, Robert L. Plackett, Applied Statistics 24 (2): 193–202, 1975.

Learning Mixtures of Plackett-Luce Models, Zhibing Zhao, Peter Piech and Lirong Xia, Proceedings of the International Conference on Machine Learning (pp. 2906-2914), 2016.

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