We study robust versions of pricing problems where customers choose products according to a generalized extreme value (GEV) choice model, and the choice parameters are not known exactly but lie in an uncertainty set. We show that, when the robust problem is unconstrained and the price sensitivity parameters are homogeneous, the robust optimal prices have a constant markup over products, and we provide formulas that allow to compute this constant markup by bisection. We further show that, in the case that the price sensitivity parameters are only homogeneous in each partition of the products, under the assumption that the choice probability generating function and the uncertainty set are partition-wise separable, a robust solution will have a constant markup in each subset, and this constant-markup vector can be found efficiently by convex optimization. We provide numerical results to illustrate the advantages of our robust approach in protecting from bad scenarios. Our results hold for convex and bounded uncertainty sets,} and for any arbitrary GEV model, including the multinomial logit, nested or cross-nested logit.
翻译:我们研究各种价格问题的稳健版本,即客户根据普遍极端价值选择模式选择产品,而选择参数则不完全为人所知,而是存在于一个不确定因素中。我们表明,当稳健的问题不受限制,价格敏感度参数是同质的,稳健的最佳价格对产品有恒定的标记,我们提供公式,以便用两部分来计算这种恒定的标记。我们进一步表明,如果价格敏感度参数在产品的每个分区中都是均匀的,假设选择概率生成功能和不确定性集是分区性可分离的,那么稳健的解决方案在每个子集中都会有一个恒定的标记,而这种恒定标记矢量则可以通过凝固优化来有效地找到。我们提供了数字结果,以说明我们稳健的方法在防止坏景象方面的优势。我们的结果可以用于连接和捆绑的不确定性组,}以及任意的GEV模型,包括多数值的对线、嵌或交叉断的对线。