Multe: Multichoice Logit Estimation

A Python library for estimating discrete choice models where agents can select either single alternatives or unordered pairs of alternatives.

This implements the model as described in Ophem, H.V., Stam, P. and Praag, B.V., 1999. Multichoice Logit: Modeling Incomplete Preference Rankings of Classical Concerts. Journal of Business & Economic Statistics, 17(1), pp.117-128.

Installation

Install from PyPI (once published):

pip install multe

Or install from source for development:

git clone https://github.com/tmonk/multe.git
cd multe
pip install -e .

Quick Start

from multe import MultichoiceLogit, simulate_data

# Generate synthetic data
X, y_single, y_dual, true_beta = simulate_data(N=1000, J=4, K=3, seed=42)

# Fit model
model = MultichoiceLogit(num_alternatives=4, num_covariates=3)
model.fit(X, y_single, y_dual)

# Access fitted coefficients
print(model.coef_)  # Shape: (J-1, K) = (3, 3)

See examples/simple_fit_example.py for a complete example, or examples/basic_example.py for advanced usage.

Model

Agents choose from J alternatives. The utility for agent i and alternative j is:

U_ij = V_ij + epsilon_ij
V_ij = X_i * beta_j
epsilon_ij ~ Gumbel(0, 1)

Each agent makes either:

• Single choice: Select the alternative with maximum utility
• Dual choice: Select the top two alternatives as an unordered pair

The mixing probability between single and dual choices is determined by the parameter mix_ratio in simulation.

Probabilities

Single choice (multinomial logit):

P(choose j) = exp(V_ij) / sum_k exp(V_ik)

Dual choice (inclusion-exclusion principle):

P(choose {s,t}) = P(U_s > max_k≠s U_k ∪ U_t > max_k≠t U_k)

Identification

The model fixes beta_0 = 0 for identification, so it estimates (J-1) × K parameters.

Data Format

• X: Covariates (N, K)
• y_single: Binary matrix (N, J) where y_single[i,j]=1 if agent i chose alternative j
• y_dual: Binary tensor (N, J, J) where y_dual[i,s,t]=1 if agent i chose pair {s,t} with s<t

Each agent must have exactly one choice (one entry in either y_single or y_dual).

Performance

Fully vectorized implementation for fast estimation and simulation:

Estimation:

• ~10ms per 1000 observations (optimization)
• Example: N=10,000, J=5, K=4 completes in ~0.1s

Recommended optimizer: L-BFGS-B

Run benchmarks: python examples/benchmark.py

API Reference

MultichoiceLogit(num_alternatives, num_covariates)

Model class with methods:

• fit(X, y_single, y_dual, method='L-BFGS-B') - Fit model using MLE (recommended)
  • Returns self with fitted coef_ attribute
  • Stores optimization details in optimization_result_
• neg_log_likelihood(flat_beta, X, y_single, y_dual) - Negative log-likelihood
• gradient(flat_beta, X, y_single, y_dual) - Analytical gradient
• compute_standard_errors(flat_beta, X, y_single, y_dual) - Standard errors

simulate_data(N, J, K, true_beta=None, mix_ratio=0.5, seed=42)

Generate synthetic data following the RUM framework.

Returns: X, y_single, y_dual, true_beta