We present prior robust algorithms for a large class of resource allocation problems where requests arrive one-by-one (online), drawn independently from an unknown distribution at every step. We design a single algorithm that, for every possible underlying distribution, obtains a 1−ϵ fraction of the profit obtained by an algorithm that knows the entire request sequence ahead of time. The factor ϵ approaches 0 when no single request consumes/contributes a significant fraction of the global consumption/contribution by all requests together. We show that the tradeoff we obtain here that determines how fast ϵ approaches 0, is near optimal: We give a nearly matching lower bound showing that the tradeoff cannot be improved much beyond what we obtain. Going beyond the model of a static underlying distribution, we introduce the adversarial stochastic input model, where an adversary, possibly in an adaptive manner, controls the distributions from which the requests are drawn at each step. Placing no restriction on the adversary, we design an algorithm that obtains a 1−ϵ fraction of the optimal profit obtainable w.r.t. the worst distribution in the adversarial sequence. Further, if the algorithm is given one number per distribution, namely the optimal profit possible for each of the adversary’s distribution, then we design an algorithm that achieves a 1−ϵ fraction of the weighted average of the optimal profit of each distribution the adversary picks. In the offline setting we give a fast algorithm to solve very large linear programs (LPs) with both packing and covering constraints. We give algorithms to approximately solve (within a factor of 1+ϵ) the mixed packing-covering problem with O (γ m log ( n /δ)/ϵ 2 ) oracle calls where the constraint matrix of this LP has dimension n × m , the success probability of the algorithm is 1−δ, and γ quantifies how significant a single request is when compared to the sum total of all requests. We discuss implications of our results to several special cases including online combinatorial auctions, network routing, and the adwords problem.

Unit demand auctions power today's search and native ad marketplaces. Traditional implementations make an extreme "separability" assumption: the relative value of any two ad slots is the same for all advertisers. Under this assumption, the optimal assignment problem can be conveniently solved simply by sorting; without it, efficient allocation requires solving a full-blown weighted matching problem. Motivated by prior work and our own empirical evidence against separability, we abandon that assumption and tackle the algorithmic problems of assignment and pricing for general unit demand ad auctions. Instead of computing prices directly, we take a novel approach and compute bidders' full allocation curves---complete mappings from each agent's bid space to their allocation under the optimal assignment function---from which it is trivial to compute most prices of interest, like those of the Generalized Second Price (GSP) or Vickrey-Clarke-Groves (VCG) auctions. Remarkably, we show that these full allocation curves (and therefore prices) can be computed in the same asymptotic runtime required to compute the optimal matching alone.

The generalized second price (GSP) auction has served as the core selling mechanism for sponsored search ads for over a decade. However, recent trends expanding the set of allowed ad formats---to include a variety of sizes, decorations, and other distinguishing features---have raised critical problems for GSP-based platforms. Alternatives such as the Vickrey-Clarke-Groves (VCG) auction raise different complications because they fundamentally change the way prices are computed. In this paper we report on our efforts to redesign a search ad selling system from the ground up in this new context, proposing a mechanism that optimizes an entire slate of ads globally and computes prices that achieve properties analogous to those held by GSP in the original, simpler setting of uniform ads. A careful algorithmic coupling of allocation-optimization and pricing-computation allows our auction to operate within the strict timing constraints inherent in real-time ad auctions. We report performance results of the auction in Yahoo's Gemini Search platform.

Nearly fifteen years ago, Google unveiled the generalized second price (GSP) auction. By all theoretical accounts including their own [Varian 14], this was the wrong auction --- the Vickrey-Clarke-Groves (VCG) auction would have been the proper choice --- yet GSP has succeeded spectacularly.

The first-price auction is popular in practice for its simplicity and transparency. Moreover, its potential virtues grow in complex settings where incentive compatible auctions may generate little or no revenue. Unfortunately, the first-price auction is poorly understood in theory because equilibrium is not a priori a credible predictor of bidder behavior.

Truthfulness is fragile and demanding. It is oftentimes computationally harder than solving the original problem. Even worse, truthfulness can be utterly destroyed by small uncertainties in a mechanism's outcome. One obstacle is that truthful payments depend on outcomes other than the one realized, such as the lengths of non-shortest-paths in a shortest-path auction. Single-call mechanisms are a powerful tool that circumvents this obstacle --- they implicitly charge truthful payments, guaranteeing truthfulness in expectation using only the outcome realized by the mechanism. The cost of such truthfulness is a trade-off between the expected quality of the outcome and the risk of large payments.

We present algorithms for a class of resource allocation problems both in the online setting with stochastic input and in the offline setting. This class of problems contains many interesting special cases such as the Adwords problem. In the online setting we introduce a new distributional model called the adversarial stochastic input model, which is a generalization of the i.i.d model with unknown distributions, where the distributions can change over time. In this model we give a 1-O(ε) approximation algorithm for the resource allocation problem, with almost the weakest possible assumption: the ratio of the maximum amount of resource consumed by any single request to the total capacity of the resource, and the ratio of the profit contributed by any single request to the optimal profit is at most (ε2/log(1/ε)2)/(log n + log (1/ε)) where n is the number of resources available. There are instances where this ratio is #949;2/log n such that no randomized algorithm can have a competitive ratio of 1-o(ε) even in the i.i.d model. The upper bound on ratio that we require improves on the previous upper-bound for the i.i.d case by a factor of n.

The convexity assumptions required for the Arrow-Debreu theorem are reasonable and realistic for preferences; however, they are highly problematic for production because they rule out economies of scale. We take a complexity-theoretic look at economies with non-convex production. It is known that in such markets equilibrium prices may not exist; we show that it is an intractable problem to achieve Pareto efficiency, the fundamental objective achieved by equilibrium prices. The same is true for core efficiency or any one of an array of concepts of stability, with the degree of intractability ranging from F Δ2P-completeness to PSPACE-hardness. We also identify a novel phenomenon that we call complexity equilibrium in which agents quiesce, not because there is no way for any one of group of them to improve their situation, but because discovering the changes necessary for (individual or group) improvement is intractable. In fact, we exhibit a somewhat natural distribution of economies that has an average-case hard complexity equilibrium.