福利最大化的多维动态定价
Multidimensional Dynamic Pricing for Welfare Maximization
摘要：
我们研究的问题，卖方动态定价$d$不同类型的不可分割的商品，当面对的在线到达的单位需求的买家独立绘制从一个未知的分布。这些商品的供应并不有限，但只能以有限的速度生产，而且生产成本很高。卖方只观察每天购买的一捆货物，而不观察买方的估价函数。我们的主要结果是一个动态的定价算法优化福利（包括卖方的生产成本），运行在时间和多个回合是多项式的$d$和近似参数。我们能够做到这一点，尽管事实上，（i）价格反应函数是不连续的，甚至它的分数松弛是一个非凹函数的价格，（ii）福利是不可观察的卖方。我们得出这个结果作为一个应用程序的一般技术优化福利超过\emph{divisible}商品，这是独立的利益。当购买者具有强凹的、Hölder连续的价值函数时，我们给出了$d$可分商品的一般多项式时间动态定价方法.我们能够将这种技术应用于单位需求购买者的设定，尽管在该设定中商品是不可分的，并且单位需求估值的自然分数松弛不是强凹的。为了应用我们的一般技术，我们引入了一种新的价格随机化过程，其效果隐含诱导买家“正规化”他们的估值与强凹函数。最后，我们还将我们的结果扩展到一个有限的供应设置，其中每个好的副本的数量不能补充。
英文摘要：
We study the problem of a seller dynamically pricing $d$ distinct types of indivisible goods, when faced with the online arrival of unit-demand buyers drawn independently from an unknown distribution. The goods are not in limited supply, but can only be produced at a limited rate and are costly to produce. The seller observes only the bundle of goods purchased at each day, but nothing else about the buyer's valuation function. Our main result is a dynamic pricing algorithm for optimizing welfare (including the seller's cost of production) that runs in time and a number of rounds that are polynomial in $d$ and the approximation parameter. We are able to do this despite the fact that (i) the price-response function is not continuous, and even its fractional relaxation is a non-concave function of the prices, and (ii) the welfare is not observable to the seller. We derive this result as an application of a general technique for optimizing welfare over \emph{divisible} goods, which is of independent interest. When buyers have strongly concave, Hölder continuous valuation functions over $d$ divisible goods, we give a general polynomial time dynamic pricing technique. We are able to apply this technique to the setting of unit demand buyers despite the fact that in that setting the goods are not divisible, and the natural fractional relaxation of a unit demand valuation is not strongly concave. In order to apply our general technique, we introduce a novel price randomization procedure which has the effect of implicitly inducing buyers to "regularize" their valuations with a strongly concave function. Finally, we also extend our results to a limited-supply setting in which the number of copies of each good cannot be replenished.
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