{"id":392486,"date":"2017-04-17T00:00:38","date_gmt":"2017-04-17T07:00:38","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=392486"},"modified":"2018-10-16T20:12:15","modified_gmt":"2018-10-17T03:12:15","slug":"optimal-multi-unit-mechanisms-private-demands","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/optimal-multi-unit-mechanisms-private-demands\/","title":{"rendered":"Optimal Multi-Unit Mechanisms with Private Demands"},"content":{"rendered":"<p>In the multi-unit pricing problem, multiple units of a single item are for sale. A buyer\u2019s\u00a0valuation for n units of the item is <strong><em>v<\/em> min<em>{n, d}<\/em><\/strong>, where the per unit valuation <strong><em>v<\/em><\/strong> and the capacity\u00a0<strong><em>d<\/em><\/strong> are private information of the buyer. We consider this problem in the Bayesian setting,\u00a0where the pair <strong>(<em>v, d<\/em>)<\/strong> is drawn jointly from a given probability distribution. In the unlimited\u00a0supply setting, the optimal (revenue maximizing) mechanism is a pricing problem, i.e., it is\u00a0a menu of lotteries. In this paper, we show that under a natural regularity condition on the\u00a0probability distributions, which we call decreasing marginal revenue, the optimal pricing is in\u00a0fact deterministic. It is a price curve, offering <strong><em>i<\/em><\/strong> units of the item for a price of <strong><em>p<sub>i<\/sub><\/em><\/strong>, for every\u00a0integer <em>i<\/em>. Further, we show that the revenue as a function of the prices <strong><em>p<sub>i<\/sub><\/em><\/strong>\u00a0is a concave function,\u00a0which implies that the optimum price curve can be found in polynomial time. This gives a rare\u00a0example of a natural multi-parameter setting where we can show such a clean characterization\u00a0of the optimal mechanism. We also give a more detailed characterization of the optimal prices\u00a0for the case where there are only two possible demands.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the multi-unit pricing problem, multiple units of a single item are for sale. A buyer\u2019s\u00a0valuation for n units of the item is v min{n, d}, where the per unit valuation v and the capacity\u00a0d are private information of the buyer. We consider this problem in the Bayesian setting,\u00a0where the pair (v, d) is drawn 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