We consider the online problem in which an intermediary trades identical items with a sequence of n buyers and n sellers, each of unit demand. We assume that the values of the traders are selected by an adversary and the sequence is randomly permuted. We give competitive algorithms for two objectives: welfare and gain-from-trade.
Recent work on online auctions for digital goods has explored the role of optimal stopping theory -- particularly secretary problems -- in the design of approximately optimal online mechanisms. This work generally assumes that the size of the market (number of bidders) is known a priori, but that the mechanism designer has no knowledge of the distribution of bid values. However, in many real-world applications (such as online ticket sales), the opposite is true: the seller has distributional knowledge of the bid values (e.g., via the history of past transactions in the market), but there is uncertainty about market size. Adopting the perspective of automated mechanism design, introduced by Conitzer and Sandholm, we develop algorithms that compute an optimal, or approximately optimal, online auction mechanism given access to this distributional knowledge. Our main results are twofold. First, we show that when the seller does not know the market size, no constant-approximation to the optimum efficiency or revenue is achievable in the worst case, even under the very strong assumption that bid values are i.i.d. samples from a distribution known to the seller. Second, we show that when the seller has distributional knowledge of the market size as well as the bid values, one can do well in several senses. Perhaps most interestingly, by combining dynamic programming with prophet inequalities (a technique from optimal stopping theory) we are able to design and analyze online mechanisms which are temporally strategyproof (even with respect to arrival and departure times) and approximately efficiency (revenue)-maximizing. In exploring the interplay between automated mechanism design and prophet inequalities, we prove new prophet inequalities motivated by the auction setting. We study the classic mathematical economics problem of 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 [20]. Unfortunately, these single parameter optimal mechanisms are impractical and rarely employed [1], and furthermore the underlying economic theory fails to generalize to the important, relevant, and unsolved multi-dimensional setting (i.e., where each agent's preference is given by multiple values for each of the multiple services available) [25]. In contrast to the theory of optimal mechanisms we develop a theory of sequential posted price mechanisms, where agents in sequence are offered take-it-or-leave-it prices. We prove that these mechanisms are approximately optimal in single-dimensional settings. These posted-price mechanisms avoid many of the properties of optimal mechanisms that make the latter impractical. Furthermore, these mechanisms generalize naturally to multi-dimensional settings where they give the first known approximations to the elusive optimal multi-dimensional mechanism design problem. In particular, we solve multi-dimensional multi-unit auction problems and generalizations to matroid feasibility constraints. The constant approximations we obtain range from 1.5 to 8. For all but one case, our posted price sequences can be computed in polynomial time. This work can be viewed as an extension and improvement of the single-agent algorithmic pricing work of [9] to the setting of multiple agents where the designer has combinatorial feasibility constraints on which agents can simultaneously obtain each service. 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 optimal welfare. Our result is constructive: given black-box access to a combinatorial auction algorithm A, sample access to the prior distribution, and appropriate query access to the sampled valuations, one can compute, in polytime, prices that guarantee at least half of the expected welfare of A. As a corollary, we obtain the first polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular combinatorial auctions, given access to demand query oracles. Our results also extend to valuations with complements, where the approximation factor degrades linearly with the level of complementarity. We study a limited-supply online auction problem, in which an auctioneer has k goods to sell and bidders arrive and depart dynamically. We suppose that agent valuations are drawn independently from some unknown distribution and construct an adaptive auction that is nevertheless value- andtime-strategy proof. For the k=1 problem we have a strategyproof variant on the classic secretary problem. We present a 4-competitive (e-competitive) strategyproof online algorithm with respect to offline Vickrey for revenue (efficiency). We also show (in a model that slightly generalizes the assumption of independent valuations) that no mechanism can be better than 3 2-competitive (2-competitive) for revenue (efficiency). Our general approach considers a learning phase followed by an accepting phase, and is careful to handle incentive issues for agents that span the two phases. We extend to the k›1 case, by deriving strategyproof mechanisms which are constant-competitive for revenue and efficiency. Finally, we present some strategyproof competitive algorithms for the case in which adversary uses a distribution known to the mechanism. We study online multi-unit auctions in which each agent’s private type consists of the agent’s arrival and departure times, valuation function and budget. Similarly to secretary settings, the different attributes of the agents’ types are determined by an adversary, but the arrival process is random. We establish a general framework for devising truthful random sampling mechanisms for online multi-unit settings with budgeted agents. We demonstrate the applicability of our framework by applying it to different objective functions (revenue and liquid welfare), and a range of assumptions about the agents’ valuations (additive or general) and the items’ nature (divisible or indivisible). Our main result is the design of mechanisms for additive bidders with budget constraints that extract a constant fraction of the optimal revenue, for divisible and indivisible items (under a standard large market assumption). We also show a mechanism that extracts a constant fraction of the optimal liquid welfare for general valuations over divisible items. In the classical secretary problem, a set S of numbers is presented to an online algorithm in random order. At any time the algorithm may stop and choose the current element, and the goal is to maximize the probability of choosing the largest element in the set. We study a variation in which the algorithm is allowed to choose k elements, and the goal is to maximize their sum. We present an algorithm whose competitive ratio is 1-O(√1 k). To our knowledge, this is the first algorithm whose competitive ratio approaches 1 as k ← ∞. As an application we solve an open problem in the theory of online auction mechanisms.