Latent Variable Models for Tied Preferences

• The paper introduces a latent variable model that naturally incorporates ties using geometric formulations and latent utility scores.
• It leverages extensions of the Plackett-Luce and Bradley-Terry models along with semi-parametric adjustments to accurately capture tie probabilities.
• This explicit treatment of ties mitigates bias in preference estimation, benefiting applications in social choice, sports ranking, and reinforcement learning from human feedback.

A latent variable model for preferences with ties provides a generative probabilistic framework for analyzing rank-ordered or paired comparison data that accommodate the empirical reality of ties—situations in which two or more items are judged as equally preferred or “tied.” These models are foundational in statistics, ranking, and machine learning, supporting consistent estimation, inference, and prediction in settings as varied as social choice, sports ranking, and human feedback for machine learning. Key modern approaches include extensions of Plackett-Luce and Bradley-Terry models, geometric latent variable constructions, and recent semi-parametric and strength-dependent frameworks, each with distinct inferential properties and computational methodologies.

1. Latent Variable Formulations for Preferences with Ties

Latent variable models for preferences posit that each item $k$ is associated with a latent score or utility reflecting its underlying “strength,” and that observed choices or rankings are noisy manifestations of the ordering of these latent variables. Ties arise when discrete or continuous random processes associated with distinct items yield equal outcomes.

The Generalized Plackett-Luce (GPL) model represents a discrete counterpart to classical Plackett-Luce by replacing continuous exponential utilities with geometric latent variables. For observed ranking $i$ with possible ties, and $K$ total items, each item's latent variable $W_{ik}$ is an independent draw:

$$W_{ik} \sim \mathrm{Geom}(\theta_k), \quad \Pr(W_{ik}=w) = (1-\theta_k)^{w-1}\theta_k, \quad 0 < \theta_k \leq 1$$

Sorting $\{W_{ik}\}$ produces the ranking, with tied values occurring with positive probability due to the discrete nature of the geometric distribution. Stage-wise, at each rank selection, items with the minimal $W_{ik}$ are “selected,” and ties are observed when multiple items share this minimum. This construction automatically accommodates ties without additional parameters or ad hoc modifications; tie probability is a direct consequence of the latent random variable structure (Henderson, 2022).

In paired comparison contexts, latent utility models for ties generalize the classical binary frameworks. For example, the Bradley-Terry model with ties (BTT) introduces a tie-parameter $\theta \geq 1$ to model the probability of indifference:

$$\begin{aligned} P(i \succ j) &= \frac{s_i}{s_i + \theta s_j} \ P(j \succ i) &= \frac{s_j}{\theta s_i + s_j} \ P(i = j) &= \frac{(\theta^2 - 1) s_i s_j}{(s_i + \theta s_j)(\theta s_i + s_j)}, \quad s_k = \exp(u_k) \end{aligned}$$

Here, increasing $\theta$ raises the tie probability; for $i$0, ties are impossible and the model reduces to standard Bradley-Terry (Liu et al., 2024).

Other generalizations, such as the Davidson and Rao-Kupper models, further parameterize tie-propensity, permitting explicit treatment of ties through distinct model likelihoods and latent-utility constructs (Chen et al., 2024).

2. Likelihoods and Closed-form Probability Functions

Latent variable models with ties support closed-form expressions for the probability of observing a particular ranking or paired outcome involving ties. In the GPL framework for rank-ordered data, the full likelihood of a sample $i$1 is expressed as a product over observed events, with the probability of each stage's selection or tie derived from the pairwise comparison of geometric variables:

$i$2

$i$3

Likelihoods for the entire data set are then products of such stage-wise and pairwise terms (Henderson, 2022).

Analogously, paired comparison tie models (BTT, Davidson, and Rao-Kupper) admit likelihoods as sums or products of probabilities over wins, losses, and ties, given latent utilities and a tie parameter:

$i$4

where $i$5 count wins, losses, and ties for pair $i$6 respectively (Liu et al., 2024).

3. Inference: EM, Gibbs Sampling, and Optimization Algorithms

Direct likelihood maximization is supported by tractable EM and data augmentation schemes. In the GPL model, one augments observed data with the set of latent minima $i$7, permitting factorization and closed-form updates:

• E-step: Compute $i$8 for appropriate $i$9
• M-step: Closed-form update for $K$0 (under Beta priors), e.g.,

$K$1

A fully-conjugate Gibbs sampler alternates draws of latent $K$2 and updates of $K$3 using Beta posterior conditionals (Henderson, 2022).

For pairwise tie models, maximization is typically by block relaxation or minorization-maximization, possibly with latent half-wins in the EM step. Tie parameters ($K$4 or equivalent) are then optimized conditional on current latent utility estimates (Liu et al., 2024).

In neural preference settings (e.g., DPO with ties), the loss functions for Rao-Kupper or Davidson models extend binary log-likelihoods to include tie terms, and one computes gradients with respect to the model parameters to enable SGD-based end-to-end optimization. The inclusion of ties imposes a regularization-like effect, constraining preference margins to remain near zero for tied pairs and improving robust discrimination capacity (Chen et al., 2024).

4. Structural Extensions: Strength-dependent and Semi-parametric Models

Recent work addresses limitations of classical parametric tie models by introducing strength-dependent tie propensities and semi-parametric frameworks. In strength-dependent models, tie probability increases with the average “strength” of the compared items. For competitors $K$5 with strengths $K$6, the probability of a tie is:

$K$7

Parameters $K$8 capture higher tie propensities in high-strength matches; analogous terms account for order effects (e.g., home advantage), possibly also modulated by competence level. These models, fit by maximum likelihood or Bayesian MCMC, capture empirical regularities from high-level competitive domains such as chess, and significantly outperform constant-tie baseline models (Glickman, 30 May 2025).

The quasi-likelihood approach based on the Kemeny covariance coefficient $K$9 offers an alternative semi-parametric direction that is complete with respect to ties. It defines a “weak-order” score matrix accommodating ties (assigned $W_{ik}$0 if $W_{ik}$1), and yields a U-statistic estimator $W_{ik}$2. Optimization is by maximization of a strictly concave $W_{ik}$3 likelihood. Formal equivalence to both Bradley-Terry and Thurstone models is established, with analytic standard errors, Wald and likelihood-ratio inference, and Edgeworth-corrected expansions available (Hurley, 30 Dec 2025).

5. Implications of Ignoring Ties: Bias and Performance

Estimating preference models without explicit tie accommodation leads to systematic bias in inferred preference strengths. For the BTT model, Liu et al. show that when ties are ignored and standard Bradley-Terry is fit, the estimator for preference difference $W_{ik}$4 is attenuated relative to the true difference $W_{ik}$5:

$W_{ik}$6

As the true tie parameter $W_{ik}$7 increases, this bias grows, and precision in preference strength recovery is compromised. Empirical evaluation on synthetic and LLM-generated data confirms that loss functions which absorb tie information (e.g., BTT, Rao-Kupper, Davidson) achieve substantially reduced bias and superior generalization, especially as the proportion of tied labels increases (Liu et al., 2024, Chen et al., 2024).

Concretely, in reinforcement learning from human feedback (RLHF) and neural evaluation settings, naïvely including ties in classical Bradley-Terry-based optimization degrades downstream performance, while models with explicit tie likelihoods retain or improve both task metrics and regularization (as measured by KL divergence to a reference policy) (Chen et al., 2024).

6. Practical Applications and Computational Considerations

Latent variable models with ties have been instrumental across domains:

• Social choice and voting: Rank aggregation, where tied preferences are ubiquitous.
• Sports analytics: Ranking competitors, modeling outcome probabilities including draws, and quantifying home/first-move advantage.
• Natural language processing: Reward learning from human feedback, where annotators frequently express indifference; RLHF and Direct Preference Optimization (DPO) benefit from tie-aware loss functions.
• Meta-ranking and rank-correlation: Semi-parametric approaches offer robust inferential frameworks for both strict and weak (tied) orderings.

All leading models (GPL, BTT, strength-dependent, semi-parametric) admit scalable inference; latent-variable augmentation, EM-style updates, and analytical or stochastic gradient computation underpin practical estimation, with tie handling introducing negligible additional computational burden.

The choice of model should be informed by data type (complete rankings, paired comparisons, continuous or discrete outcomes), empirical tie frequency, interpretability, and desired inference granularity.

7. Connections, Equivalences, and Model Universality

Fundamental equivalences exist between tied-preference models:

• For minimal tie propensity, all models reduce to their no-tie classical limits (Plackett-Luce, Bradley-Terry).
• The weak-order score-based U-statistic model is formally equivalent to a linear Bradley-Terry or Thurstone latent-difference structure under appropriate monotone link transformations (Hurley, 30 Dec 2025).
• Strength-dependent and order-effect models subsume the Davidson and Rao-Kupper models as limiting cases.
• Empirical and theoretical analysis confirms that proper treatment of ties is essential for unbiased inference, efficient use of data, and correct uncertainty quantification in preference modeling.

The general paradigm—modeling observed (possibly tied) rankings as noisy functions of latent variables with explicit tie mechanisms—serves as a unifying foundation for theory and practice in statistical ranking, comparative judgment, and machine learning with preferences (Henderson, 2022, Chen et al., 2024, Liu et al., 2024, Glickman, 30 May 2025, Hurley, 30 Dec 2025).