{"id":392525,"date":"2017-06-10T00:00:17","date_gmt":"2017-06-10T07:00:17","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=392525"},"modified":"2018-10-16T20:12:17","modified_gmt":"2018-10-17T03:12:17","slug":"multidimensional-dynamic-pricing-welfare-maximization","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/multidimensional-dynamic-pricing-welfare-maximization\/","title":{"rendered":"Multidimensional Dynamic Pricing for Welfare Maximization"},"content":{"rendered":"<p>We study the problem of a seller dynamically pricing <em>d<\/em> distinct types of indivisible goods,\u00a0when faced with the online arrival of unit-demand buyers drawn independently from an unknown<br \/>\ndistribution. The goods are not in limited supply, but can only be produced at a limited\u00a0rate and are costly to produce. The seller observes only the bundle of goods purchased at each<br \/>\nday, but nothing else about the buyer\u2019s valuation function. Our main result is a dynamic pricing\u00a0algorithm for optimizing welfare (including the seller\u2019s cost of production) that runs in<br \/>\ntime and a number of rounds that are polynomial in <em>d<\/em> and the approximation parameter. We\u00a0are able to do this despite the fact that (i) the price-response function is not continuous, and<br \/>\neven its fractional relaxation is a non-concave function of the prices, and (ii) the welfare is not\u00a0observable to the seller.<\/p>\n<p>We derive this result as an application of a general technique for optimizing welfare over\u00a0<em>divisible<\/em> goods, which is of independent interest. When buyers have strongly concave, Holder continuous valuation functions over <em>d<\/em> divisible goods, we give a general polynomial time dynamic\u00a0pricing technique. We are able to apply this technique to the setting of unit demand<br \/>\nbuyers despite the fact that in that setting the goods are not divisible, and the natural fractional\u00a0relaxation of a unit demand valuation is not strongly concave. In order to apply our<br \/>\ngeneral technique, we introduce a novel price randomization procedure which has the effect of\u00a0implicitly inducing buyers to \u201cregularize\u201d their valuations with a strongly concave function.<br \/>\nFinally, we also extend our results to a limited-supply setting in which the number of copies of\u00a0each good cannot be replenished.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We study the problem of a seller dynamically pricing d distinct types of indivisible goods,\u00a0when 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\u00a0rate and are costly to produce. The seller observes only the bundle 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