Conferences such as FUN with Algorithms routinely buy goodies (e.g., t-shirts, coffee mugs, etc) for their attendees. Often, said goodies come in different types, varying by color or design, and organizers need to decide how many goodies of each type to buy. We study the problem of buying optimal amounts of each type under a simple model of preferences by the attendees: they are indifferent to the types but want to be able to choose between more than one type of goodies at the time of their arrival. The indifference of attendees suggests that the optimal policy is to buy roughly equal amounts for every goodie type. Despite how intuitive this conjecture sounds, we show that this simple model of assortment optimization is quite rich, and even though we make progress towards proving the conjecture (e.g., we succeed when the number of goodie types is 2 or 3), the general case with K types remains open. We also present asymptotic results and computer simulations, and finally, to motivate further progress, we offer a reward of $100usd for a full proof.

翻译：诸如FUN with Algorithms等会议通常会为参会者购买赠品（如T恤、咖啡杯等）。这些赠品常按颜色或设计分为不同类型，组织者需决定每种类型的购买数量。我们在一个简单的参会者偏好模型下研究每种类型的最优购买量问题：参会者对赠品种类无偏好，但希望到达时能在多种类型之间进行选择。这种无偏好性暗示最优策略应为每种赠品类型购买大致相等的数量。尽管这一猜想看似直观，但我们发现这个简单的组合优化模型实则相当丰富——即使我们在证明该猜想方面取得进展（例如在赠品种类数为2或3时成功证明），含有K种类型的一般情形仍未解决。我们还提出了渐近结果与计算机模拟，最后为激励进一步研究，对完整证明悬赏100美元。