Pricing for Online Resource Allocation: Intervals and Paths

Abstract

We present pricing mechanisms for several online resource allocation problems which obtain tight or nearly tight approximations to social welfare. In our settings, buyers arrive online and purchase bundles of items; buyers' values for the bundles are drawn from known distributions. This problem is closely related to the so-called prophet-inequality of Krengel and Sucheston [22] and its extensions in recent literature. Motivated by applications to cloud economics, we consider two kinds of buyer preferences. In the first, items correspond to different units of time at which a resource is available; the items are arranged in a total order and buyers desire intervals of items. The second corresponds to bandwidth allocation over a tree network; the items are edges in the network and buyers desire paths.Because buyers' preferences have complementarities in the settings we consider, recent constant-factor approximations via item prices do not apply, and indeed strong negative results are known. We develop static, anonymous bundle pricing mechanisms.For the interval preferences setting, we show that static, anonymous bundle pricings achieve a sublogarithmic competitive ratio, which is optimal (within constant factors) over the class of all online allocation algorithms, truthful or not. For the path preferences setting, we obtain a nearlytight logarithmic competitive ratio. Both of these results exhibit an exponential improvement over item pricings for these settings. Our results extend to settings where the seller has multiple copies of each item, with the competitive ratio decreasing linearly with supply. Such a gradual tradeoff between supply and the competitive ratio for welfare was previously known only for the single item prophet inequality.

Prophet Inequalities on the Intersection of a Matroid and a Graph

“…In the graph G induced by constraints (2), two agents are adjacent if their intervals overlap. For an agent t, any agents t ′ < t with I t ′ ∩ I t = ∅ must have I t ′ contain the point ℓ t , since ℓ t ′ ≤ ℓ t and the intervals are contiguous.…”

“…Our paper is most related to the existing work involving matroids [11,6,5] and XOS valuation functions [8,5]. We should mention that interval-scheduling constraints have also been studied in [10,2], where it is shown that with no assumptions on the intervals, the guarantee relative to the prophet is at most O(log log L/ log L), where L is the length of (number of items in) the longest interval. That is, with no assumptions on the intervals, a constant-factor is impossible.…”

“…There is a single server, so a set of agents S is feasible if and only if S satisfies constraints (2). Agent 1 has valuation C+T ε ε with probability ε, and valuation 0 otherwise, where C, ε > 0 are constants.…”

Market Pricing for Matroid Rank Valuations

“…Dütting et al [10,11] provided a general framework for posted-price mechanisms in Bayesian settings. Chawla et al [6] showed that static, anonymous bundle pricing mechanisms are useful when buyers' preferences have complementarities. Ezra et al [15] provided upper and lower bounds on the largest fraction of the optimal social welfare that can be guaranteed with static prices for several classes of valuations, such as submodular, XOS, or subadditive.…”

An O(log log m) prophet inequality for subadditive combinatorial auctions

“…Interest in prophet inequalities has surged recently, in part due to applications in pricing and auction design driven by the observation that the fixed threshold can be viewed as a posted price. This has lead to a line of literature studying prophet inequalities for more general instances of the allocation problem, yielding approximatelyoptimal online policies and incentive compatible auctions for increasingly general problem instances with many objects and rich classes of valuation functions (e.g., [4], [5], [6], [7], [8], [9], [10], [11], [12]).…”