We introduce the notion of non-monotone utilities, which covers a wide variety of utility functions in economic theory. We show that it is PPAD-hard to compute an approximate Arrow-Debreu market equilibrium in markets with linear and non-monotone utilities. Building on this result, we settle the long-standing open problem regarding the computation of an approximate Arrow-Debreu market equilibrium in markets with CES utilities, by proving that it is PPADcomplete when the Constant Elasticity of Substitution parameter, ρ, is any constant less than −1.

Abstract-Modern frameworks, such as Hadoop, combined with abundance of computing resources from the cloud, offer a significant opportunity to address long standing challenges in distributed processing. Infrastructure-as-a-Service clouds reduce the investment cost of renting a large data center while distributed processing frameworks are capable of efficiently harvesting the rented physical resources. Yet, the performance users get out of these resources varies greatly because the cloud hardware is shared by all users. The value for money cloud consumers achieve renders resource sharing policies a key player in both cloud performance and user satisfaction. In this paper, we employ microeconomics to direct the allotment of cloud resources for consumption in highly scalable masterworker virtual infrastructures. Our approach is developed on two premises: the cloud-consumer always has a budget and cloud physical resources are limited. Using our approach, the cloud administration is able to maximize per-user financial profit. We show that there is an equilibrium point at which our method achieves resource sharing proportional to each user's budget. Ultimately, this approach allows us to answer the question of how many resources a consumer should request from the seemingly endless pool provided by the cloud.

We introduce the notion of non-monotone utilities, which covers a wide variety of utility functions in economic theory. We show that it is PPAD-hard to compute an approximate Arrow-Debreu market equilibrium in markets with linear and non-monotone utilities. Building on this result, we settle the long-standing open problem regarding the computation of an approximate Arrow-Debreu market equilibrium in markets with CES utilities, by proving that it is PPADcomplete when the Constant Elasticity of Substitution parameter, ρ, is any constant less than −1.

We resolve the complexity of revenue-optimal deterministic auctions in the unit-demand single-buyer Bayesian setting, i.e., the optimal item pricing problem, when the buyer's values for the items are independent. We show that the problem of computing a revenue-optimal pricing can be solved in polynomial time for distributions of support size 2, and its decision version is NP-complete for distributions of support size 3. We also show that the problem remains NP-complete for the case of identical distributions. * Columbia University.consequence of the analysis is that the optimal pricing problem has the integrality property: if the values in the supports are integer then the optimal prices are also integer (though they may not belong to the support).We then proceed to show (Theorem 2) that the case in which each marginal distribution has support at most 2 can be solved in polynomial time. Indeed, by exploiting the underlying structure of the problem, we show that it suffices to consider O(n 2 ) price-vectors to compute the optimal revenue in this case.Our main result is that the problem is NP-hard, even for distributions of support 3 (Theorem 3) or distributions that are identical but have large support (Theorem 4). This answers an open problem first posed in [CHK07] and also asked in [CD11, DDT12b]. The main difficulty in the reductions stems from the fact that, for a general instance of the pricing problem, the expected revenue is a highly complex nonlinear function of the prices. The challenge is to construct an instance such that the revenue can be well-approximated by a simple function and is also general enough to encode an NP-hard problem.Previous Work. We have already mentioned the main algorithmic works for the independent distributions case with approximately-optimal revenue guarantees [CHK07, CHMS10, CD11]. On the lower bound side, Guruswami et al. [GHK + 05] and subsequently Briest [Bri08] studied the complexity of the problem when the buyer's values for the items are correlated, respectively obtaining APX-hardness and Ω(n ǫ ) inapproximability, for some constant ǫ > 0. More recently, Daskalakis, Deckelbaum and Tzamos [DDT12b] showed that the pricing problem with independent distributions is SQRT-SUM-hard when either the support values or the probabilities are irrational. We note that their reduction relies on the fact that, for certain carefully constructed instances, it is SQRT-SUM-hard to compare the revenue of two price-vectors. This has no bearing on the complexity of the problem under the standard discrete model we consider, for which the exact revenue of a price-vector can be computed efficiently.Related Work. The optimal mechanism design problem (i.e., the problem of finding a revenue-maximizing mechanism in a Bayesian setting) has received considerable attention in the CS community during the past few years. The vast majority of the work so far is algorithmic [CHK07, CHMS10, BGGM10, Ala11, DFK11, HN12, CDW12a, CDW12b], providing approximation or exact algorithms for various versions of the prob...

We study the optimal lottery problem and the optimal mechanism design problem in the setting of a single unit-demand buyer with item values drawn from independent distributions. Optimal solutions to both problems are characterized by a linear program with exponentially many variables. For the menu size complexity of the optimal lottery problem, we present an explicit, simple instance with distributions of support size 2, and show that exponentially many lotteries are required to achieve the optimal revenue. We also show that, when distributions have support size 2 and share the same high value, the simpler scheme of item pricing can achieve the same revenue as the optimal menu of lotteries. The same holds for the case of two items with support size 2 (but not necessarily the same high value). For the computational complexity of the optimal mechanism design problem, we show that unless the polynomial-time hierarchy collapses (more exactly, P NP = P #P), there is no universal efficient randomized algorithm to implement an optimal mechanism even when distributions have support size 3.