What Is the Impact of Nonrandomness on Random Choice Models?

Problem definition: This paper examines the impact of nonrandomness on random choice models and studies various operations problems under the new discrete choice models. Academic/practical relevance: The literature often assumes that the random utility components follow some independent and identically distributed distribution. This assumption is too restrictive in some real-world scenarios, because, for example, consumers may have known well about the attribute values for the product that they have repeatedly purchased. Methodology: We adopt the random utility maximization framework and characterize the choice probabilities when the utility of some alternative is deterministic. The log-likelihood function is jointly concave in the attribute coefficients under the linear utility-attribute assumption; an expectation-maximization algorithm is developed to overcome the missing data issue in estimation. Results: Surprisingly, if the utility of a particular product is deterministic, the assortment problem is still polynomial-time solvable, whereas if the utility of the no-purchase option is deterministic, the decision problem corresponding to the assortment optimization is NP-complete. We show that the price minus the reciprocal of price sensitivity is product invariant at optimality, which helps to simplify the multiproduct pricing problems. Managerial implications: Empirical study on real data shows that incorporating nonrandomness into random choice models can increase model fitting and prediction accuracy. Failure of accounting for the impact of nonrandomness may result in substantial losses.