Operations Research最新文献

We describe an efficient algorithm to compute solutions for the general two-player Blotto game on n battlefields with heterogeneous values. Whereas explicit constructions for such solutions have been limited to specific, largely symmetric or homogeneous setups, this algorithmic resolution covers the most general situation to date: a value-asymmetric game with an asymmetric budget with sufficient symmetry and homogeneity. The proposed algorithm rests on recent theoretical advances regarding Sinkhorn iterations for matrix and tensor scaling. An important case that had been out of reach of previous attempts is that of heterogeneous but symmetric battlefield values with asymmetric budgets. In this case, the Blotto game is constant-sum, so optimal solutions exist, and our algorithm samples from an $\epsilon$-optimal solution in time $\tilde{\mathcal{O}}(n^2+{\epsilon }^{-4})$, independent of budgets and battlefield values, up to some natural normalization. In the case of asymmetric values where optimal solutions need not exist but Nash equilibria do, our algorithm samples from an $\epsilon$-Nash equilibrium with similar complexity but where implicit constants depend on various parameters of the game such as battlefield values.

Funding: V. Perchet acknowledges support from the French National Research Agency (ANR) [Grant ANR-19-CE23-0026] as well as the support grant, and Investissements d’Avenir [Grant LabEx Ecodec/ANR-11-LABX-0047]. P. Rigollet is supported by the NSF [Grants IIS-1838071, DMS-2022448, and CCF-2106377].

Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2023.0049.

When a firm selects an assortment of products to offer to customers, it uses a choice model to anticipate their probability of purchasing each product. In practice, the estimation of these models is subject to statistical errors, which may lead to significantly suboptimal assortment decisions. Recent work has addressed this issue using robust optimization, where the true parameter values are assumed unknown and the firm chooses an assortment that maximizes its worst-case expected revenues over an uncertainty set of likely parameter values, thus mitigating estimation errors. In this paper, we introduce the concept of randomization into the robust assortment optimization literature. We show that the standard approach of deterministically selecting a single assortment to offer is not always optimal in the robust assortment optimization problem. Instead, the firm can improve its worst-case expected revenues by selecting an assortment randomly according to a prudently designed probability distribution. We demonstrate this potential benefit of randomization both theoretically in an abstract problem formulation as well as empirically across three popular choice models: the multinomial logit model, the Markov chain model, and the preference ranking model. We show how an optimal randomization strategy can be determined exactly and heuristically. Besides the superior in-sample performance of randomized assortments, we demonstrate improved out-of-sample performance in a data-driven setting that combines estimation with optimization. Our results suggest that more general versions of the assortment optimization problem—incorporating business constraints, more flexible choice models and/or more general uncertainty sets—tend to be more receptive to the benefits of randomization.

Funding: Z. Wang acknowledges funding from the Imperial College President’s PhD Scholarship programme. W. Wiesemann acknowledges funding from the Engineering and Physical Sciences Research Council [Grants EP/R045518/1, EP/T024712/1, and EP/W003317/1].

Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2022.0129.