Karthyek Murthy

Given data on the choices made by consumers for different offer sets, a key challenge is to develop parsimonious models that describe and predict consumer choice behavior while being amenable to prescriptive tasks, such as pricing and assortment optimization. The marginal distribution model (MDM) is one such model, which requires only the specification of marginal distributions of the random utilities. This paper aims to establish necessary and sufficient conditions for given choice data to be consistent with the MDM hypothesis inspired by the usefulness of similar characterizations for the random utility model (RUM). This endeavor leads to an exact characterization of the set of choice probabilities that the MDM can represent. Verifying the consistency of choice data with this characterization is equivalent to solving a polynomial-sized linear program. Because the analogous verification task for RUM is computationally intractable and neither of these models subsume the other, MDM is helpful in striking a balance between tractability and representational power. The characterization is then used with robust optimization for making data-driven sales and revenue predictions for new unseen assortments. When the choice data lack consistency with the MDM hypothesis, finding the best-fitting MDM choice probabilities reduces to solving a mixed-integer convex program. Numerical results using real-world data and synthetic data demonstrate that MDM exhibits competitive representational power and prediction performance compared with RUM and parametric models while being significantly faster in computation than RUM. This paper was accepted by Omar Besbes, revenue management and pricing department. Funding: This work was supported by the Singapore Ministry of Education Academic Research Fund [Grants T2EP20223-0008 and T2MOE1906]. X. Li was supported in part by the Singapore Ministry of Education Academic Research Fund Tier 1 [Grants 23-0619-P0001 and 24-0500-A0001]. The research of K. Murthy was supported by the Singapore Ministry of Education Academic Research Fund [Grants MOE2019-T2-2-163 and MOE-T2EP20223-0017]. The research of K. Natarajan was partly supported by the Singapore Ministry of Education Academic Research Fund Tier 2 [Grant MOE2019-T2-2-138]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.02287 .

Scalable and efficient importance sampling for managing tail risks As the models employed in the realm of risk analytics and optimization become increasingly sophisticated, it is crucial that risk management tools, such as variance reduction techniques, that are typically designed for stylized models on a case by case basis evolve to scale well and gain broader applicability. In the paper titled “Achieving efficiency in black-box simulation of distribution tails with self-structuring importance samplers,” the authors take a step toward this goal by introducing a novel importance sampling scheme for estimating tail risks of objectives modeled with a diverse range of tools, including linear programs, integer linear programs, feature maps specified with neural networks, etc. Instead of explicitly tailoring the change of distribution for each specific model, as conventionally done, the paper identifies an elementary transformation of the samples. This transformation, when applied alike across a wide variety of models, yields a near-optimal reduction in variance for estimating/optimizing over tail expectations.

Given data on the choices made by consumers for different offer sets, a key challenge is to develop parsimonious models that describe and predict consumer choice behavior while being amenable to prescriptive tasks such as pricing and assortment optimization. The marginal distribution model (MDM) is one such model, that requires only the specification of marginal distributions of the random utilities. This paper aims to establish necessary and sufficient conditions for given choice data to be consistent with the MDM hypothesis, inspired by the utility of similar characterizations for the random utility model (RUM). This endeavor leads to an exact characterization of the set of choice probabilities that the MDM can represent. Verifying the consistency of choice data with this characterization is equivalent to solving a polynomial-sized linear program. Since the analogous verification task for RUM is computationally intractable and neither of these models subsumes the other, MDM is helpful in striking a balance between tractability and representational power. The characterization is convenient to be used with robust optimization for making data-driven sales and revenue predictions for new unseen assortments. When the choice data lacks consistency with the MDM hypothesis, finding the best-fitting MDM choice probabilities reduces to solving a mixed integer convex program. The results extend naturally to the case where the alternatives can be grouped based on the similarity of the marginal distributions of the utilities. Numerical experiments show that MDM provides better representational power and prediction accuracy than multinominal logit and significantly better computational performance than RUM.

Given data on choices made by consumers for different assortments, a key challenge is to develop parsimonious models that describe and predict consumer choice behavior. One such choice model is the marginal distribution model (MDM), which requires only the specification of the marginal distributions of the random utilities of the alternatives to explain choice data.

This paper investigates the use of retrospective approximation solution paradigm in solving risk-averse optimization problems effectively via importance sampling (IS). While IS serves as a prominent means for tackling the large sample requirements in estimating tail risk measures such as Conditional Value at Risk (CVaR), its use in optimization problems driven by CVaR is complicated by the need to tailor the IS change of measure differently to different optimization iterates and the circularity which arises as a consequence. The proposed algorithm overcomes these challenges by employing a univariate IS transformation offering uniform variance reduction in a retrospective approximation procedure well-suited for tuning the IS parameter choice. The resulting simulation based approximation scheme enjoys both the computational efficiency bestowed by retrospective approximation and logarithmically efficient variance reduction offered by importance sampling

We consider optimal transport-based distributionally robust optimization (DRO) problems with locally strongly convex transport cost functions and affine decision rules. Under conventional convexity assumptions on the underlying loss function, we obtain structural results about the value function, the optimal policy, and the worst-case optimal transport adversarial model. These results expose a rich structure embedded in the DRO problem (e.g., strong convexity even if the non-DRO problem is not strongly convex, a suitable scaling of the Lagrangian for the DRO constraint, etc., which are crucial for the design of efficient algorithms). As a consequence of these results, one can develop efficient optimization procedures that have the same sample and iteration complexity as a natural non-DRO benchmark algorithm, such as stochastic gradient descent.

This paper deals with the problem of quantifying the impact of model misspecification when computing general expected values of interest. The methodology that we propose is applicable in great generality; in particular, we provide examples involving path-dependent expectations of stochastic processes. Our approach consists of computing bounds for the expectation of interest regardless of the probability measure used, as long as the measure lies within a prescribed tolerance measured in terms of a flexible class of distances from a suitable baseline model. These distances, based on optimal transportation between probability measures, include Wasserstein’s distances as particular cases. The proposed methodology is well suited for risk analysis and distributionally robust optimization, as we demonstrate with applications. We also discuss how to estimate the tolerance region nonparametrically using Skorokhod-type embeddings in some of these applications.