We study the problem of designing posted-price mechanisms in order to sell a single unit of a single item within a finite period of time. Motivated by real-world problems, such as, e.g., long-term rental of rooms and apartments, we assume that customers arrive online according to a Poisson process, and their valuations are drawn from an unknown distribution and discounted over time. We evaluate our mechanisms in terms of competitive ratio, measuring the worst-case ratio between their revenue and that of an optimal mechanism that knows the distribution of valuations. First, we focus on the identical valuation setting, where all the customers value the item for the same amount. In this setting, we provide a mechanism M_c that achieves the best possible competitive ratio, discussing its dependency on the parameters in the case of linear discount. Then, we switch to the random valuation setting. We show that, if we restrict the attention to distributions of valuations with a monotone hazard rate, then the competitive ratio of M_c is lower bounded by a strictly positive constant that does not depend on the distribution. Moreover, we provide another mechanism, called M_pc, which is defined by a piecewise constant pricing strategy and reaches performances comparable to those obtained with M_c. This mechanism is useful when the seller cannot change the posted price too often. Finally, we empirically evaluate the performances of our mechanisms in a number of experimental settings.
We study the revenue performance of sequential posted price mechanisms and some natural extensions, for a general setting where the valuations of the buyers are drawn from a correlated distribution. Sequential posted price mechanisms are conceptually simple mechanisms that work by proposing a take-it-or-leave-it offer to each buyer. We apply sequential posted price mechanisms to single-parameter multi-unit settings in which each buyer demands only one item and the mechanism can assign the service to at most k of the buyers. For standard sequential posted price mechanisms, we prove that with th…
We study the asymptotic average-case efficiency of static and anonymous posted prices for $n$ agents and $m(n)$ multiple identical items with $m(n)=o\left(\frac{n}{\log n}\right)$. When valuations are drawn i.i.d from some fixed continuous distribution (each valuation is a vector in $\Re_+^m$ and independence is assumed only across agents) we show: (a) for any "upper mass" distribution there exist posted prices such that the expected revenue and welfare of the auction approaches the optimal expected welfare as $n$ goes to infinity; specifically, the ratio between the expected revenue of our po…
We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in an arbitrary order, each purchasing her favorite bundle from among the unsold items at the posted prices. These mechanisms are simple, transparent and trivially dominant strategy incentive compatible (DSIC). We show that when agent preferences are fractionally subadditive (which includes all submodular functions), there always exist prices that, in expectation, obtain at least half of the optim…
We study the power and limitations of posted prices in multi-unit markets, where agents arrive sequentially in an arbitrary order. We prove upper and lower bounds on the largest fraction of the optimal social welfare that can be guaranteed with posted prices, under a range of assumptions about the designer's information and agents' valuations. Our results provide insights about the relative power of uniform and non-uniform prices, the relative difficulty of different valuation classes, and the implications of different informational assumptions. Among other results, we prove constant-factor gu…
In the quest for market mechanisms that are easy to implement, yet close to optimal, few seem as viable as posted pricing. Despite the growing body of impressive results, the performance of most posted price mechanisms however, rely crucially on price discrimination when multiple copies of a good are available. For the more general case with non-linear production costs on each good, hardly anything is known for general multi-good markets. With this in mind, we study a Bayesian setting where the seller can produce any number of copies of a good but faces convex production costs for the same, an…
We study revenue optimization pricing algorithms for repeated posted-price auctions where a seller interacts with a single strategic buyer that holds a fixed private valuation. We show that, in the case when both the seller and the buyer have the same discounting in their cumulative utilities (revenue and surplus), there exist two optimal algorithms. The first one constantly offers the Myerson price, while the second pricing proposes a "big deal": pay for all goods in advance (at the first round) or get nothing. However, when there is an imbalance between the seller and the buyer in the patien…
We consider the problem of maximizing the expected revenue from selling $k$ homogeneous goods to $n$ unit-demand buyers who arrive sequentially with independent and identically distributed valuations. In this setting the optimal posted prices are dynamic in the sense that they depend on the remaining numbers of goods and buyers. We investigate how much revenue is lost when a single static price is used instead for all buyers and goods, and prove upper bounds on the ratio between the maximum revenue from dynamic prices and that from static prices. These bounds are tight for all values of $k$ an…
We consider the problem of posting prices for unit-demand buyers if all $n$ buyers have identically distributed valuations drawn from a distribution with monotone hazard rate. We show that even with multiple items asymptotically optimal welfare can be guaranteed. Our main results apply to the case that either a buyer's value for different items are independent or that they are perfectly correlated. We give mechanisms using dynamic prices that obtain a $1 - Θ\left( \frac{1}{\log n}\right)$-fraction of the optimal social welfare in expectation. Furthermore, we devise mechanisms that only use sta…
We consider an online ad network problem in which an ad exchange auctions ad slots and intermediaries called demand side platforms (DSPs) buy these ad slots for their clients (advertisers). An intermediary represents multiple advertisers. Different types of ad slots are auctioned by the ad exchange, e.g., video ad, banner ad etc. We study repeated posted price auctions for homogeneous and heterogeneous items when there is an intermediary. In a posted price auction, the auctioneer sets a fixed reserve price. The buyer can accept the price and win the ad slot or reject the price. We analyze the …
Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here $n$ agents with subadditive valuation functions compete for the assignment of $m$ items. The goal is to find an allocation of the …
We obtain revenue guarantees for the simple pricing mechanism of a single posted price, in terms of a natural parameter of the distribution of buyers' valuations. Our revenue guarantee applies to the single item n buyers setting, with values drawn from an arbitrary joint distribution. Specifically, we show that a single price drawn from the distribution of the maximum valuation Vmax = max {V_1, V_2, ...,V_n} achieves a revenue of at least a 1/e fraction of the geometric expecation of Vmax. This generic bound is a measure of how revenue improves/degrades as a function of the concentration/sprea…
We study single-item single-unit Bayesian posted price auctions, where buyers arrive sequentially and their valuations for the item being sold depend on a random, unknown state of nature. The seller has complete knowledge of the actual state and can send signals to the buyers so as to disclose information about it. For instance, the state of nature may reflect the condition and/or some particular features of the item, which are known to the seller only. The problem faced by the seller is about how to partially disclose information about the state so as to maximize revenue. Unlike classical sig…
We consider the problem of designing a revenue-optimal mechanism in the two-item, single-buyer, unit-demand setting when the buyer's valuations, $(z_1, z_2)$, are uniformly distributed in an arbitrary rectangle $[c,c+b_1]\times[c,c+b_2]$ in the positive quadrant. We provide a complete and explicit solution for arbitrary nonnegative values of $(c,b_1,b_2)$. We identify five simple structures, each with at most five (possibly stochastic) menu items, and prove that the optimal mechanism has one of the five structures. We also characterize the optimal mechanism as a function of $b_1, b_2$, and $c$…
We study the mechanism design problem of selling $k$ items to unit-demand buyers with private valuations for the items. A buyer either participates directly in the auction or is represented by an intermediary, who represents a subset of buyers. Our goal is to design robust mechanisms that are independent of the demand structure (i.e. how the buyers are partitioned across intermediaries), and perform well under a wide variety of possible contracts between intermediaries and buyers. We first study the case of $k$ identical items where each buyer draws its private valuation for an item i.i.d. fro…
We study revenue optimization learning algorithms for repeated posted-price auctions where a seller interacts with a single strategic buyer that holds a fixed private valuation for a good and seeks to maximize his cumulative discounted surplus. For this setting, first, we propose a novel algorithm that never decreases offered prices and has a tight strategic regret bound in $Θ(\log\log T)$ under some mild assumptions on the buyer surplus discounting. This result closes the open research question on the existence of a no-regret horizon-independent weakly consistent pricing. The proposed algorit…
We study revenue optimization learning algorithms for repeated second-price auctions with reserve where a seller interacts with multiple strategic bidders each of which holds a fixed private valuation for a good and seeks to maximize his expected future cumulative discounted surplus. We propose a novel algorithm that has strategic regret upper bound of $O(\log\log T)$ for worst-case valuations. This pricing is based on our novel transformation that upgrades an algorithm designed for the setup with a single buyer to the multi-buyer case. We provide theoretical guarantees on the ability of a tra…
A large fraction of online advertisement is sold via repeated second price auctions. In these auctions, the reserve price is the main tool for the auctioneer to boost revenues. In this work, we investigate the following question: Can changing the reserve prices based on the previous bids improve the revenue of the auction, taking into account the long-term incentives and strategic behavior of the bidders? We show that if the distribution of the valuations is known and satisfies the standard regularity assumptions, then the optimal mechanism has a constant reserve. However, when there is uncert…
We study revenue maximization in multi-item multi-bidder auctions under the natural item-independence assumption - a classical problem in Multi-Dimensional Bayesian Mechanism Design. One of the biggest challenges in this area is developing algorithms to compute (approximately) optimal mechanisms that are not brute-force in the size of the bidder type space, which is usually exponential in the number of items in multi-item auctions. Unfortunately, such algorithms were only known for basic settings of our problem when bidders have unit-demand [CHMS10,CMS15] or additive valuations [Yao15]. In thi…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Consider the seller's problem of finding optimal prices for her $n$ (divisible) goods when faced with a set of $m$ consumers, given that she can only observe their purchased bundles at posted prices, i.e., revealed preferences. We study both social welfare and profit maximization with revealed preferences. Although social welfare maximization is a seemingly non-convex optimization problem in prices, we show that (i) it can be reduced to a dual convex optimization problem in prices, and (ii) the revealed preferences can be interpreted as supergradients of the concave conjugate of valuation, wit…
The existing literature on optimal auctions focuses on optimizing the expected revenue of the seller, and is appropriate for risk-neutral sellers. In this paper, we identify good mechanisms for risk-averse sellers. As is standard in the economics literature, we model the risk-aversion of a seller by endowing the seller with a monotone concave utility function. We then seek robust mechanisms that are approximately optimal for all sellers, no matter what their levels of risk-aversion are. We have two main results for multi-unit auctions with unit-demand bidders whose valuations are drawn i.i.d. …
We study the performance of anonymous posted-price selling mechanisms for a standard Bayesian auction setting, where $n$ bidders have i.i.d. valuations for a single item. We show that for the natural class of Monotone Hazard Rate (MHR) distributions, offering the same, take-it-or-leave-it price to all bidders can achieve an (asymptotically) optimal revenue. In particular, the approximation ratio is shown to be $1+O(\ln \ln n/\ln n)$, matched by a tight lower bound for the case of exponential distributions. This improves upon the previously best-known upper bound of $e/(e-1)\approx 1.58$ for th…
