A Convex Optimization Approach for Computing Correlated Choice Probabilities With Many Alternatives

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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, Natarajan and Teo (IEEE TAC, 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

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