Designing Networks with Good Equilibria under Uncertainty

We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of k terminal vertices or players need to establish connectivity with the root. The social optimum is the Minimum Steiner Tree. We are interested in situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying metric but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the designer's goal is to choose a single, universal protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is: to what extent can prior knowledge of the underlying metric help in the design? We first demonstrate that there exist graphs (outerplanar) where knowledge of the underlying metric can dramatically improve the performance of good network design. Then, in our main technical result, we show that there exist graph metrics, for which knowing the underlying metric does not help and any universal protocol has PoA of $\Omega(\log k)$, which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey Theory in high dimensional hypercubes. Then we switch to the stochastic model, where each player is independently activated. We show that there exists a randomized ordered protocol that achieves constant PoA. By using standard derandomization techniques, we produce a deterministic ordered protocol with constant PoA.Comment: This version has additional results about stochastic inpu

A deterministic truthful PTAS for scheduling related machines

Scheduling on related machines ($Q||C_{\max}$) is one of the most important problems in the field of Algorithmic Mechanism Design. Each machine is controlled by a selfish agent and her valuation can be expressed via a single parameter, her {\em speed}. In contrast to other similar problems, Archer and Tardos \cite{AT01} showed that an algorithm that minimizes the makespan can be truthfully implemented, although in exponential time. On the other hand, if we leave out the game-theoretic issues, the complexity of the problem has been completely settled -- the problem is strongly NP-hard, while there exists a PTAS \cite{HS88,ES04}. This problem is the most well studied in single-parameter algorithmic mechanism design. It gives an excellent ground to explore the boundary between truthfulness and efficient computation. Since the work of Archer and Tardos, quite a lot of deterministic and randomized mechanisms have been suggested. Recently, a breakthrough result \cite{DDDR08} showed that a randomized truthful PTAS exists. On the other hand, for the deterministic case, the best known approximation factor is 2.8 \cite{Kov05,Kov07}. It has been a major open question whether there exists a deterministic truthful PTAS, or whether truthfulness has an essential, negative impact on the computational complexity of the problem. In this paper we give a definitive answer to this important question by providing a truthful {\em deterministic} PTAS

On the Efficiency of the Proportional Allocation Mechanism for Divisible Resources

We study the efficiency of the proportional allocation mechanism, that is widely used to allocate divisible resources. Each agent submits a bid for each divisible resource and receives a fraction proportional to her bids. We quantify the inefficiency of Nash equilibria by studying the Price of Anarchy (PoA) of the induced game under complete and incomplete information. When agents' valuations are concave, we show that the Bayesian Nash equilibria can be arbitrarily inefficient, in contrast to the well-known 4/3 bound for pure equilibria. Next, we upper bound the PoA over Bayesian equilibria by 2 when agents' valuations are subadditive, generalizing and strengthening previous bounds on lattice submodular valuations. Furthermore, we show that this bound is tight and cannot be improved by any simple or scale-free mechanism. Then we switch to settings with budget constraints, and we show an improved upper bound on the PoA over coarse-correlated equilibria. Finally, we prove that the PoA is exactly 2 for pure equilibria in the polyhedral environment.Comment: To appear in SAGT 201

Tight Bounds for the Price of Anarchy of Simultaneous First Price Auctions

We study the Price of Anarchy of simultaneous first-price auctions for buyers with submodular and subadditive valuations. The current best upper bounds for the Bayesian Price of Anarchy of these auctions are e/(e-1) [Syrgkanis and Tardos 2013] and 2 [Feldman et al. 2013], respectively. We provide matching lower bounds for both cases even for the case of full information and for mixed Nash equilibria via an explicit construction. We present an alternative proof of the upper bound of e/(e-1) for first-price auctions with fractionally subadditive valuations which reveals the worst-case price distribution, that is used as a building block for the matching lower bound construction. We generalize our results to a general class of item bidding auctions that we call bid-dependent auctions (including first-price auctions and all-pay auctions) where the winner is always the highest bidder and each bidder's payment depends only on his own bid. Finally, we apply our techniques to discriminatory price multi-unit auctions. We complement the results of [de Keijzer et al. 2013] for the case of subadditive valuations, by providing a matching lower bound of 2. For the case of submodular valuations, we provide a lower bound of 1.109. For the same class of valuations, we were able to reproduce the upper bound of e/(e-1) using our non-smooth approach.Comment: 37 pages, 5 figures, ACM Transactions on Economics and Computatio

Groundwater investigation in Paphos region

Cyprus is a semi-arid island. Its water resources rely on winter rainfall which supply the impounding reservoirs constructed on the dry river courses and replenish the groundwater resources within the river gravels or the plain aquifers. water requirements for domestic use and irrigation have been increasing considerably for the last ten years. The potential of the conventional water resources have been developed according to techno-economic factors. Also, the unconventional sources (treated domestic effluents and desalinating sea water) are receiving particular attention to support water requirements throughout the island. The current study deals partly with the calcarenite aquifer of the Paphos Coastal Plain. It has been investigated whether this resource is offered for an integrated exploitation program. In addition the domestic effluent of the Paphos urban zone is considered as a promising resource for providing reliable and continuous quantities for irrigation use after treatment. [Continues.

Truthful Allocation in Graphs and Hypergraphs

We study truthful mechanisms for allocation problems in graphs, both for the minimization (i.e., scheduling) and maximization (i.e., auctions) setting. The minimization problem is a special case of the well-studied unrelated machines scheduling problem, in which every given task can be executed only by two pre-specified machines in the case of graphs or a given subset of machines in the case of hypergraphs. This corresponds to a multigraph whose nodes are the machines and its hyperedges are the tasks. This class of problems belongs to multidimensional mechanism design, for which there are no known general mechanisms other than the VCG and its generalization to affine minimizers. We propose a new class of mechanisms that are truthful and have significantly better performance than affine minimizers in many settings. Specifically, we provide upper and lower bounds for truthful mechanisms for general multigraphs, as well as special classes of graphs such as stars, trees, planar graphs, k-degenerate graphs, and graphs of a given treewidth. We also consider the objective of minimizing or maximizing the L^p-norm of the values of the players, a generalization of the makespan minimization that corresponds to p = ?, and extend the results to any p > 0