In this paper, a rather general online problem called dynamic resource allocation with capacity constraints (DRACC) is introduced and studied in the realm of posted price mechanisms. This problem subsumes several applications of stateful pricing, including but not limited to posted prices for online job scheduling and matching over a dynamic bipartite graph. As the existing online learning techniques do not yield vanishing-regret mechanisms for this problem, we develop a novel online learning framework defined over deterministic Markov decision processes with dynamic state transition and reward functions. We then prove that if the Markov decision process is guaranteed to admit an oracle that can simulate any given policy from any initial state with bounded loss -- a condition that is satisfied in the DRACC problem -- then the online learning problem can be solved with vanishing regret. Our proof technique is based on a reduction to online learning with switching cost, in which an online decision maker incurs an extra cost every time she switches from one arm to another. We formally demonstrate this connection and further show how DRACC can be used in our proposed applications of stateful pricing.
We consider the problem of budget feasible mechanism design proposed by Singer (2010), but in a Bayesian setting. A principal has a public value for hiring a subset of the agents and a budget, while the agents have private costs for being hired. We consider both additive and submodular value functions of the principal. We show that there are simple, practical, ex post budget balanced posted pricing mechanisms that approximate the value obtained by the Bayesian optimal mechanism that is budget balanced only in expectation. A main motivating application for this work is the crowdsourcing large p…
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…
Crowd sensing is a new paradigm which leverages a large number of sensor-equipped mobile phones to collect sensing data. To attract more participants to provide good quality, bidding mechanisms that solicit the Vickrey-Clarke-Groves (VCG) mechanism and its variants are natural fits for crowd sensing applications in mobile social networks. However, in practical continuous crowd sensing applications, where bids cannot be solicited and only posted pricing mechanisms can be implemented. Obviously, these mechanisms for continuous crowd sensing are not applicable. To tackle the issue, we propose a c…
We consider job scheduling settings, with multiple machines, where jobs arrive online and choose a machine selfishly so as to minimize their cost. Our objective is the classic makespan minimization objective, which corresponds to the completion time of the last job to complete. The incentives of the selfish jobs may lead to poor performance. To reconcile the differing objectives, we introduce posted machine prices. The selfish job seeks to minimize the sum of its completion time on the machine and the posted price for the machine. Prices may be static (i.e., set once and for all before any arr…
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 iden…
We consider the classical mathematical economics problem of {\em Bayesian optimal mechanism design} where a principal aims to optimize expected revenue when allocating resources to self-interested agents with preferences drawn from a known distribution. In single-parameter settings (i.e., where each agent's preference is given by a single private value for being served and zero for not being served) this problem is solved [Myerson '81]. Unfortunately, these single parameter optimal mechanisms are impractical and rarely employed [Ausubel and Milgrom '06], and furthermore the underlying economic…
We study online resource allocation in a cloud computing platform, through a posted pricing mechanism: The cloud provider publishes a unit price for each resource type, which may vary over time; upon arrival at the cloud system, a cloud user either takes the current prices, renting resources to execute its job, or refuses the prices without running its job there. We design pricing functions based on the current resource utilization ratios, in a wide array of demand-supply relationships and resource occupation durations, and prove worst-case competitive ratios of the pricing functions in terms …
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…
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 design algorithms for computing approximately revenue-maximizing {\em sequential posted-pricing mechanisms (SPM)} in $K$-unit auctions, in a standard Bayesian model. A seller has $K$ copies of an item to sell, and there are $n$ buyers, each interested in only one copy, who have some value for the item. The seller must post a price for each buyer, the buyers arrive in a sequence enforced by the seller, and a buyer buys the item if its value exceeds the price posted to it. The seller does not know the values of the buyers, but have Bayesian information about them. An SPM specifies the orderin…
We study revenue maximization through sequential posted-price (SPP) mechanisms in single-dimensional settings with $n$ buyers and independent but not necessarily identical value distributions. We construct the SPP mechanisms by considering the best of two simple pricing rules: one that imitates the revenue optimal mchanism, namely the Myersonian mechanism, via the taxation principle and the other that posts a uniform price. Our pricing rules are rather generalizable and yield the first improvement over long-established approximation factors in several settings. We design factor-revealing mathe…
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…
Online markets are a part of everyday life, and their rules are governed by algorithms. Assuming participants are inherently self-interested, well designed rules can help to increase social welfare. Many algorithms for online markets are based on prices: the seller is responsible for posting prices while buyers make purchases which are most profitable given the posted prices. To make adjustments to the market the seller is allowed to update prices at certain timepoints. Posted prices are an intuitive way to design a market. Despite the fact that each buyer acts selfishly, the seller's goal is …
We consider dynamic pricing schemes in online settings where selfish agents generate online events. Previous work on online mechanisms has dealt almost entirely with the goal of maximizing social welfare or revenue in an auction settings. This paper deals with quite general settings and minimizing social costs. We show that appropriately computed posted prices allow one to achieve essentially the same performance as the best online algorithm. This holds in a wide variety of settings. Unlike online algorithms that learn about the event, and then make enforceable decisions, prices are posted wit…
This paper considers Bayesian revenue maximization in the $k$-unit setting, where a monopolist seller has $k$ copies of an indivisible item and faces $n$ unit-demand buyers (whose value distributions can be non-identical). Four basic mechanisms among others have been widely employed in practice and widely studied in the literature: {\sf Myerson Auction}, {\sf Sequential Posted-Pricing}, {\sf $(k + 1)$-th Price Auction with Anonymous Reserve}, and {\sf Anonymous Pricing}. Regarding a pair of mechanisms, we investigate the largest possible ratio between the two revenues (a.k.a.\ the revenue gap)…
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 present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing lite…
The common way to optimize auction and pricing systems is to set aside a small fraction of the traffic to run experiments. This leads to the question: how can we learn the most with the smallest amount of data? For truthful auctions, this is the \emph{sample complexity} problem. For posted price auctions, we no longer have access to samples. Instead, the algorithm is allowed to choose a price $p_t$; then for a fresh sample $v_t \sim \mathcal{D}$ we learn the sign $s_t = sign(p_t - v_t) \in \{-1,+1\}$. How many pricing queries are needed to estimate a given parameter of the underlying distribut…
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…
Dynamic pricing schemes were introduced as an alternative to posted-price mechanisms. In contrast to static models, the dynamic setting allows to update the prices between buyer-arrivals based on the remaining sets of items and buyers, and so it is capable of maximizing social welfare without the need for a central coordinator. In this paper, we study the existence of optimal dynamic pricing schemes in combinatorial markets. In particular, we concentrate on multi-demand valuations, a natural extension of unit-demand valuations. The proposed approach is based on computing an optimal dual soluti…
We consider a fundamental problem in microeconomics: selling a single item to a number of potential buyers, whose values are drawn from known independent and regular (not necessarily identical) distributions. There are four widely-used and widely-studied mechanisms in the literature: {\sf Myerson Auction}~({\sf OPT}), {\sf Sequential Posted-Pricing}~({\sf SPM}), {\sf Second-Price Auction with Anonymous Reserve}~({\sf AR}), and {\sf Anonymous Pricing}~({\sf AP}). {\sf OPT} is revenue-optimal but complicated, which also experiences several issues in practice such as fairness; {\sf AP} is the sim…
Shanghai Jiao Tong University（上海交通大学）；University of Texas at Dallas（德克萨斯大学达拉斯分校）
The society's insatiable appetites for personal data are driving the emergency of data markets, allowing data consumers to launch customized queries over the datasets collected by a data broker from data owners. In this paper, we study how the data broker can maximize her cumulative revenue by posting reasonable prices for sequential queries. We thus propose a contextual dynamic pricing mechanism with the reserve price constraint, which features the properties of ellipsoid for efficient online optimization, and can support linear and non-linear market value models with uncertainty. In particul…
New York University（纽约大学）；Massachusetts Institute of Technology（麻省理工学院）；University of Florida（佛罗里达大学）
The prevalence of e-commerce has made detailed customers' personal information readily accessible to retailers, and this information has been widely used in pricing decisions. When involving personalized information, how to protect the privacy of such information becomes a critical issue in practice. In this paper, we consider a dynamic pricing problem over $T$ time periods with an \emph{unknown} demand function of posted price and personalized information. At each time $t$, the retailer observes an arriving customer's personal information and offers a price. The customer then makes the purcha…
Stochastic Online Fisher Markets: Static Pricing Limits and Adaptive Enhancements
Institute for Computational and Mathematical Engineering, Stanford University（斯坦福大学计算与数学工程研究所）；Department of Management Science and Engineering, Stanford University（斯坦福大学管理科学与工程学院）
Microsoft Research（微软研究院）；The University of Hong Kong（香港大学）；Stanford University（斯坦福大学）
We consider revenue maximization in online auction/pricing problems. A seller sells an identical item in each period to a new buyer, or a new set of buyers. For the online posted pricing problem, we show regret bounds that scale with the best fixed price, rather than the range of the values. We also show regret bounds that are almost scale free, and match the offline sample complexity, when comparing to a benchmark that requires a lower bound on the market share. These results are obtained by generalizing the classical learning from experts and multi-armed bandit problems to their multi-scale …
In this paper, we present the first approximation algorithms for the problem of designing revenue optimal Bayesian incentive compatible auctions when there are multiple (heterogeneous) items and when bidders can have arbitrary demand and budget constraints. Our mechanisms are surprisingly simple: We show that a sequential all-pay mechanism is a 4 approximation to the revenue of the optimal ex-interim truthful mechanism with discrete correlated type space for each bidder. We also show that a sequential posted price mechanism is a O(1) approximation to the revenue of the optimal ex-post truthful…
Optimal stopping theory is a powerful tool for analyzing scenarios such as online auctions in which we generally require optimizing an objective function over the space of stopping rules for an allocation process under uncertainty. Perhaps the most classic problems of stopping theory are the prophet inequality problem and the secretary problem. The classical prophet inequality states that by choosing the same threshold OPT/2 for every step, one can achieve the tight competitive ratio of 0.5. On the other hand, for the basic secretary problem, the optimal strategy achieves the tight competitive…
