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A convex optimization approach for computing correlated choice probabilities with many alternatives

A convex optimization approach for computing correlated choice probabilities with many alternatives
A convex optimization approach for computing correlated choice probabilities with many alternatives
A popular discrete choice model that incorporates correlation information is the multinomial probit (MNP) model where the random utilities of the alternatives are chosen from a multivariate normal distribution. Computing the choice probabilities is challenging in the MNP model when the number of alternatives is large. Mishra et al. (IEEE Transactions on Automatic Control, 2012) have proposed a semidefinite optimization approach to compute choice probabilities for the distribution of the random utilities that maximizes expected agent utility given only the mean, variance, and covariance information. Their model is referred to as the cross moment (CMM) model. Computing the choice probabilities with many alternatives is challenging in the CMM model, since one needs to solve large-scale semidefinite programs. We develop a simpler formulation as a representative agent model by maximizing over the choice probabilities in the unit simplex where the objective function is the sum of the expected utilities and a strongly concave perturbation function. By characterizing the perturbation function for the CMM model and its gradient, we develop a simple first-order gradient method with inexact line search to compute choice probabilities. We establish local linear convergence of this algorithm under mild assumptions on the choice probabilities. An implication of our results is that inverting the choice probabilities to compute the mean utilities is straightforward given any positive-definite covariance matrix. Numerical experiments show that this method can compute choice probabilities for a large number of alternatives within a reasonable amount of time while explicitly capturing the correlation information. Comparisons with simulation methods for MNP and semidefinite programming methods for CMM indicate the efficacy of the method.
Choice probability, optimization, optimization algorithms, stochastic systems
0018-9286
190-205
Ahipasaoglu, Selin Damla
d69f1b80-5c05-4d50-82df-c13b87b02687
Li, Xiaobo
e7ea954f-9fae-4d52-b41f-ab93ecd883b3
Natarajan, Karthik
f06e68cf-288e-4280-8cdb-5b5d7d397ce0
Ahipasaoglu, Selin Damla
d69f1b80-5c05-4d50-82df-c13b87b02687
Li, Xiaobo
e7ea954f-9fae-4d52-b41f-ab93ecd883b3
Natarajan, Karthik
f06e68cf-288e-4280-8cdb-5b5d7d397ce0

Ahipasaoglu, Selin Damla, Li, Xiaobo and Natarajan, Karthik (2019) A convex optimization approach for computing correlated choice probabilities with many alternatives. IEEE Transactions on Automatic Control, 64 (1), 190-205. (doi:10.1109/TAC.2018.2875673).

Record type: Article

Abstract

A popular discrete choice model that incorporates correlation information is the multinomial probit (MNP) model where the random utilities of the alternatives are chosen from a multivariate normal distribution. Computing the choice probabilities is challenging in the MNP model when the number of alternatives is large. Mishra et al. (IEEE Transactions on Automatic Control, 2012) have proposed a semidefinite optimization approach to compute choice probabilities for the distribution of the random utilities that maximizes expected agent utility given only the mean, variance, and covariance information. Their model is referred to as the cross moment (CMM) model. Computing the choice probabilities with many alternatives is challenging in the CMM model, since one needs to solve large-scale semidefinite programs. We develop a simpler formulation as a representative agent model by maximizing over the choice probabilities in the unit simplex where the objective function is the sum of the expected utilities and a strongly concave perturbation function. By characterizing the perturbation function for the CMM model and its gradient, we develop a simple first-order gradient method with inexact line search to compute choice probabilities. We establish local linear convergence of this algorithm under mild assumptions on the choice probabilities. An implication of our results is that inverting the choice probabilities to compute the mean utilities is straightforward given any positive-definite covariance matrix. Numerical experiments show that this method can compute choice probabilities for a large number of alternatives within a reasonable amount of time while explicitly capturing the correlation information. Comparisons with simulation methods for MNP and semidefinite programming methods for CMM indicate the efficacy of the method.

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More information

Accepted/In Press date: 9 February 2018
Published date: January 2019
Keywords: Choice probability, optimization, optimization algorithms, stochastic systems

Identifiers

Local EPrints ID: 443194
URI: http://eprints.soton.ac.uk/id/eprint/443194
ISSN: 0018-9286
PURE UUID: 53d3bc65-3a97-43ca-8093-f26da0bc02e3
ORCID for Selin Damla Ahipasaoglu: ORCID iD orcid.org/0000-0003-1371-315X

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Date deposited: 13 Aug 2020 16:38
Last modified: 17 Mar 2024 04:03

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Contributors

Author: Xiaobo Li
Author: Karthik Natarajan

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