From pricing to prophets, and back!

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Abstract

In this work we prove that designing PPMs is equivalent to finding stopping rules for prophets. This extends the connection that any prophet type inequality can be turned into a PPM with the same approximation guarantee (Hajiaghayi et al. 2007; Chawla et al. 2010). Our reduction is robust under multiple settings including matroid feasibility constraints, or different arrival orderings. One fundamental observation implied by this result is that designing PPMs in general is equally hard from an approximation perspective to designing PPMs when the valuations are regular.

Introduction

In the last few years online sales have been moving from an auction format, to posted price formats [15] and the basic reason for this trend switch seems to be that posted price mechanisms (PPM) are much simpler than optimal auctions, yet efficient enough. The way these mechanisms work is as follows. Suppose a seller has an item to sell. Customers arrive one at a time and the seller proposes to each customer a take-it-or-leave-it price. The first customer accepting the offer pays the price and takes the item. These types of mechanisms are flexible and adapt well to different scenarios; their simplicity and the fact that strategic behavior vanishes make them quite suitable for many applications [9]. Of course, PPMs are suboptimal and therefore the study of their approximation guarantees – where the benchmark is that given by the optimal Myerson’s auction [28] – has been an extremely active area in the last decade, in particular in the computer science community.

Hajiaghayi et al. [20] and Chawla et al. [9] establish an interesting connection between (revenue maximizing) PPMs and prophet inequalities, a problem arising in optimal stopping theory. Here a gambler is faced to a sequence of random variables and has to pick a stopping time so that the expected value he gets is as close as possible to the expectation of the maximum of all random variables, interpreted as what a prophet, who knows the realizations in advance, could get. They implicitly show that any prophet type inequality can be turned into a PPM with the same approximation guarantee. This is obtained by noting that a PPM for revenue maximization can be seen as a (threshold) stopping rule for the gambler, but on the virtual values space, and later identify these virtual thresholds with prices. As a consequence, the follow up work in the field concentrated on prophet inequities and then applied the obtained results to sequential PPMs.

In this work we fill a gap in this line of research by proving the converse of the latter result, namely, that any posted price mechanism can be turned into a prophet type inequality with the same approximation guarantee. The core of the result is a method to go back from virtual values to arbitrary distributions which may find applications beyond the scope of this paper. This result amounts not only to apply approximation guarantees from prophet inequalities to PPMs, but also to carry over the lower bounds. We observe that through our reduction we can improve the best known lower bound for sequential PPMs (in which the arrival order is either random or selected by the seller) in the single item case, the k-uniform matroid case, the general matroid case, and the general downward-closed family case.

The recent survey by Lucier [26] is an excellent starting point in the area, where many variants of PPMs are described. For the specific scenario where only one item must be allocated, some pricing setting studied include anonymous (the offered price is the same for all customers) [2], [9], [13], static (the possibly different prices to offer do not evolve as the mechanism progresses) [11], [16], and Order-Oblivious (the order in which agents arrive can be chosen by an adaptive adversary) [9].

Furthermore, PPMs may be used when selling multiple items or with constraints on the subsets of served customers. Typical side constraints include matroids constraints [23], [33], downward-closed systems [5], [29], combinatorial prophet inequalities [8], [30], combinatorial auctions [1], [17], and polymatroids constraints [14]. Attention has also been payed to settings with limited information or prior-independent, where the designer must learn the distribution in order to run the mechanism [3], [4], [7], [10], [12], [27].

For fixed n>1, let X1,,Xn be non-negative, independent random variables and Tn their set of stopping rules. A classic result of Krengel and Sucheston, and Gairing [24], [25] asserts that E(max{X1,,Xn})2sup{E(Xt):tTn}, and that 2 is the best possible bound. The study of this type of inequalities, known as prophet inequalities, was initiated by Gilbert and Mosteller [18] and attracted a lot of attention in the eighties [21], [22], [23], [31], [32]. In particular Samuel-Cahn [32] noted that rather than looking at the set of all stopping rules one can (quite naturally) only look at threshold stopping rules in which the decision to stop depends on whether the value of the currently observed random variable is above a certain threshold.

The main insight we derive is a valuation mapping lemma stating that for any distribution F there is another distribution G whose virtual value distributes according to F. It is surprising that this basic result was missing from the auction theory literature and we believe that it may prove useful in settings beyond PPMs.

Our result is robust to different settings. It applies to random, adversarial, or best possible orders, as well as when there are multiple items and constraints on the allowed allocation sets. As already mentioned before, the sufficiency condition of the theorem is a known fact and although it has never appeared explicitly, it is implicit in previous work [9], [20]. The necessary condition, however, is novel and not obvious. The main difficulty comes from taking an arbitrary distribution in the prophet inequality problem and mapping it back to a PPM. Here is where the valuation mapping lemma, that holds for arbitrary distributions, comes into play. Consider the operator that picks an arbitrary probability distribution over the nonnegative reals and returns the distribution of the ironed virtual valuation function. The Valuation Mapping Lemma states that this operator is surjective over the space of distributions. Interestingly, the lemma gives an explicit construction so we can easily interpret the thresholds as prices in the PPM.

A remarkable feature of the Valuation Mapping Lemma is that when mapping a distribution F into another distribution G whose virtual value follows F, G turns out to be regular (i.e., it has a monotone non-decreasing virtual value). Although in principle there may be many functions G satisfying the statement of the lemma, we can identify one explicitly with this appealing property. Together with our main theorem these imply that the posted price problem can be reduced to a prophet inequality problem, which can in turn be reduced to a posted price problem with regular distributions. Therefore, designing PPMs in general is equally hard from an approximation perspective to designing PPMs when the valuations are regular.

Another consequence of our results is that we can translate all known upper and lower bounds from PPMs into prophet inequalities and back. One example which we will further analyze in Section 4 is the case of sequential posted price mechanisms (SPM, [9]). The current best known lower bound for this setting is π21.253 [6]. This is also the best known when the feasibility constraint is a general matroid, and even the intersection of two matroids. Our result implies an improvement on this bound to 1.341 by using the lower bound for the i.i.d. prophet inequality designed by Hill and Kertz [21]. Although our results are presented in the context of single-parameter mechanism design, they can be generalized to multi-parameter settings [9].

In Section 2 we introduce formally the online selection problem and the auction problem. In Section 3 we prove our main result – formally stated in Theorem 9 –, that is, the reduction from PPM to prophet inequalities. In Section 4 we show the improved lower bound for SPMs in more detail.

Section snippets

Preliminaries

Online selection problem. An instance of this problem corresponds to a tuple (X,F,T), where X is the ground set of n elements and each set in T2X is called feasible selection. For each xX there is a random variable wx, called weight, distributed according to Fx with compact support contained in R+, and F={Fx:xX}. We assume them to be independent. The random variables are presented in an order σ:[n]X, and an algorithm for the problem has to decide whether to select or not an element of X

Reduction overview.

Consider an instance (X,F,T) for the optimal stopping problem, and suppose we have access to a single-parameter PPM M that provides a guarantee over the ground set X and feasibility constraints T. If we were able to find valuation distributions G={Gx:xX} such that ϕGx+(vx) has distribution Fx, where vx has distribution Gx, then we could feed the mechanism M with the instance (X,G,T) of a multi-item auction problem, using the same order σ in which the elements of the ground set X are output in

Implications

A direct consequence of Theorem 9 and the Valuation Mapping Lemma is that we obtain lower bounds for the guarantees of PPMs by considering lower bound instances of the online selection problem. We improve the previous known lower bounds for SPM when constraints are on the form of downward closed families, from logn(3loglogn) to logn(2loglogn) [9], [29], and in the k-uniform matroid setting from 1.253 to 1.341 [6], [21] (this holds for general matroids or even intersection of matroids).

Acknowledgment

Dana Pizarro was supported by CONICYT /DoctoradoNacional/2016-21161440, Chile.

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