Data-Driven Optimization of Personalized Reserve Prices Mahsa Derakhshan University of Maryland Negin Golrezaei MIT Renato Paes Leme Google Research Abstract We study the problem of computing data-driven personalized reserve prices in eager second price auctions without having any assumption on valuation distributions. Here, the input is a data-set that contains the submitted bids of n buyers in a set of auctions and the problem is to return personalized reserve prices r that maximize the revenue earned on these auctions by running eager second price auctions with reserve r. For this problem, which is known to be NP-hard, we present a novel LP formulation and a rounding procedure which achieves a (1 + 2( √ 2 − 1)e √ 2-2 ) -1 ≈ 0.684-approximation. This improves over the 1 2 -approximation algorithm due to Roughgarden and Wang. We show that our analysis is tight for this rounding procedure. We also bound the integrality gap of the LP, which shows that it is impossible to design an algorithm that yields an approximation factor larger than 0.828 with respect to this LP. 1 Introduction Second price (Vickrey) auctions with reserves have been prevalent in many marketplaces such as online advertising markets [GLMN17, PLPV16, CS14]. A key parameter of this auction format is its reserve price, which is the minimum price at which the seller is willing to sell an item. While there are empirical and theoretical evidence that highlight the significance of setting personalized reserve prices for the buyers to maximize the revenue [EOS07, OS11, BGL + 18], we do not have a full understanding of how to optimize reserve prices. This problem is only fully solved under the assumption that buyers’ valuation distributions are i.i.d. and regular, where these assumptions fail to hold in practice [GLMN17, CLMN14]. We study the problem of optimizing personalized reserve prices in second price auctions when the buyer valuations can be correlated. There are two different ways that personalized reserve prices can be applied in the second price auctions: lazy and eager [DRY15]. In the lazy version, we first determine the potential winner and then apply the reserve prices. In the eager version, we first apply the reserve prices and then determine the winner. In this work, we focus on optimizing eager reserve prices because (i) while the optimal lazy reserve prices can be computed exactly in polynomial time, they have worse revenue performance both in theory and practice, and (ii) eager reserves perform better in terms of social efficiency for similar revenue levels [PLPV16]. To optimize the eager reserve prices, we take a data-driven approach as suggested in the litera- ture [PLPV16, RW16]. The input in this setting is a history of the buyers’ submitted bids/valuations over multiple runs of an auction and the goal, roughly speaking, is to set a personalized reserve price r b for each buyer b such that the total revenue obtained on the same data-set according to these reserve prices is maximized (see Section 2 for the formal definition). The optimal data-driven reserve prices solve an offline optimization problem, i.e., given a data- 1 arXiv:1905.01526v3 [cs.GT] 30 Mar 2020