We consider a market setting of agents with additive valuations over heterogeneous divisible resources. Agents are assigned a budget of tokens (possibly unequal budgets) they can use to obtain resources; leftover tokens are worthless. We show how to support any Pareto efficient allocation in equilibrium, using anonymous resource prices and agent specific budgets. We also give computationally efficient algorithms for those tasks. In particular, this allows us to support the Rawlsian max-min allocation.

A strong equilibrium (Aumann 1959) is a pure Nash equilibrium which is resilient to deviations by coalitions. We define the strong price of anarchy to be the ratio of the worst case strong equilibrium to the social optimum. In contrast to the traditional price of anarchy, which quantifies the loss incurred due to both selfishness and lack of coordination, the strong price of anarchy isolates the loss originated from selfishness from that obtained due to lack of coordination. We study the strong price of anarchy in two settings, one of job scheduling and the other of network creation. In the job scheduling game we show that for unrelated machines the strong price of anarchy can be bounded as a function of the number of machines and the size of the coalition. For the network creation game we show that the strong price of anarchy is at most 2. In both cases we show that a strong equilibrium always exists, except for a well defined subset of network creation games.

We consider the task of designing truthful mechanisms for single parameter agents. We prove a general sufficient condition for truthfulness when each agent's valuation function for each possible outcome is a one-dimensional function of its type, continuous everywhere and differentiable almost everywhere. For certain types of natural valuation functions, our condition is also necessary. Our condition extends both the Mirrlees-Spence condition [25, 17], applicable only for differentiable real allocations, and Archer and Tardos' single parameter characterization [4], which assumes an agent's valuation is linear in its type.We demonstrate the simplicity of testing our condition by showing that classical criteria for truthfulness in combinatorial problems such as auctions and machine scheduling can be derived from our condition. In addition, we use our condition to derive results for new single parameter problems, which have not been previously analyzed.We also consider combinatorial problems where the true types of agents affect the valuation of each other, such as in machine scheduling with selfish jobs. In such cases there are only degenerate dominant strategy mechanisms. We show that the same condition can be used to design mechanisms which are ex-post truthful, meaning that the outcome where all agents cooperate and report their true type is a Nash equilibrium. We demonstrate the power of this condition by applying it on the problem of machine scheduling with strategic job owners, previously presented in [5]. We give a constant approximation ratio algorithms for the original problem and to the double setting where both jobs and machines are strategic.

We consider the task of designing truthful mechanisms for single parameter agents. We prove a general sufficient condition for truthfulness when the valuation function of the agents of any outcome is one dimensional and continuous. For certain types of natural valuation functions, our condition is also necessary. One of the main advantages of our characterization is that it provides a computationally efficient method for testing whether a given algorithm admits a truthful payment scheme. We then demonstrate the simplicity of testing our condition by showing that classical criteria for truthfulness in combinatorial problems such as auctions and machine scheduling can be derived from our condition. In addition, we use our condition to derive results for new single parameter problems, which have not been previously analyzed.We also consider combinatorial problems where the true types of agents affect the valuation of each other, such as in load balancing. In such cases there are no efficient dominant strategy mechanisms. We show that the same condition can be used to design mechanisms which are ex-post truthful, meaning that the outcome where all agents cooperate and report their true type is a Nash equilibrium. We demonstrate the power of this condition by applying it on the problem of machine scheduling with strategic job owners, which was not addressed before. We prove upper and lower bounds for ex-post truthful mechanisms for this problem.

We focus on the online problem of queue management in networks providing differentiated services. As in DiffServ, packets are divided into two priority groups. Low priority packets are assigned the value of 1 and high priority packets are assigned the value of α > 1. The goal is to maximize the total value of packets that are sent. Restricted to FIFO queues, the packets must be sent by the order of their arrival, however we are allowed to preempt packets from the queue.Several deterministic online algorithms for this model have been presented in previous papers. Currently, the best online algorithm known for this problem has a competitive ratio of 1.304 for the worst case α [17]. In this work we consider randomized online algorithms. Our main result is an online policy that outperforms any deterministic policy and achieves a competitive ratio of 1.25, by using a single random bit. This result is lower than the deterministic lower bound of 1.281 [19]. We then derive a general lower bound for randomized algorithms of 1.197.A natural extension of this model is to assign arbitrary packet values to the input packets. Currently, the best competitive ratio achieved for a deterministic policy is 1.75 [7]. We present a randomized comparison based online policy with the same competitive ratio. Since no deterministic comparison based policy is known to have a competitive ratio better than 2, we believe that this result demonstrates the potential of using randomization to outperform deterministic policies, as in the two value model.

We consider the problem of scheduling jobs on related machines owned by selfish agents. We provide a 5-approximation deterministic truthful mechanism, the first deterministic truthful result for the problem. Previously, Archer and Tardos showed a 2-approximation randomized mechanism which is truthful in expectation only (a weaker notion of truthfulness). In case the number of machines is constant, we provide a deterministic Fully Polynomial-Time Approximation Scheme (FPTAS) and a suitable payment scheme that yields a truthful mechanism for the problem. This result, which is based on converting FPTAS to monotone FPTAS, improves a previous result of Auletta et al., who showed a (4 + ε)-approximation truthful mechanism.

In a combinatorial auction k different items are sold to n bidders, where the objective of the seller is to maximize the revenue. The main difficulty to find an optimal allocation is due to the fact that the valuation function of each bidder for bundles of items is not necessarily an additive function over the items. An auction with budget constraints is a common special case where bidders generally have additive valuations, yet they have a limit on their maximal valuation. Auctions with budget constraints were analyzed by Lehmann, Lehmann and Nisan [11], as part of a wider class of auctions, where they have shown that maximizing the revenue is NP-hard, and presented a greedy 2-approximation algorithm. In this paper we present exact and approximate algorithms for auctions with budget constraints. We present a randomized algorithm with an approximation ratio of e/e-1≅ 1.582, which can be derandomized. We analyze the special case where all bidders have the same budget constraint, and show an algorithm whose approximation ratio is between 1.3837 and 1.3951. We also present an FPTAS for the case of a constant number of bidders.

We consider the online problem of active queue management. In our model, the input is a sequence of packets with values v ∈ [1,α] that arrive to a queue that can hold up to B packets. Specifically, we consider a FIFO non-preemptive queue, where any packet that is accepted into the queue must be sent, and packets are sent by the order of arrival. The benefit of a scheduling policy, on a given input, is the sum of values of the scheduled packets. Our aim is to find an online policy that maximizes its benefit compared to the optimal offline solution. Previous work proved that no constant competitive ratio exists for this problem, showing a lower bound of ln(α) + 1 for any online policy. An upper bound of e⌈ln(α) ⌉ was proved for a few online policies. In this paper we suggest and analyze a RED-like online policy with a competitive ratio that matches the lower bound up to an additive constant proving an upper bound of ln(α) + 2 + O(ln 2(α)/B). For large values of α, we prove that no policy whose decisions are based only on the number of packets in the queue and the value of the arriving packet, has a competitive ratio lower than ln(α) + 2 - ε, for any constant ε > 0.

We consider packet scheduling in a network providing differentiated services, where each packet is assigned a value. We study various queueing models for supporting QoS (Quality of Service). In the nonpreemptive model, packets accepted to the queue will be transmitted eventually and cannot be dropped. The FIFO preemptive model allows packets accepted to the queue to be preempted (dropped) prior to their departure, while ensuring that transmitted packets are sent in the order of arrival. In the bounded delay model, packets must be transmitted before a certain deadline, otherwise it is lost (while transmission ordering is allowed to be arbitrary). In all models the goal of the buffer policy is to maximize the total value of the accepted packets.Let α be the ratio between the maximal and minimal value. For the non-preemptive model we derive a Θ(log &alpha) competitive ratio, both exhibiting a buffer policy and a general lower bound. For the interesting case of two distinct values, we give an 2α--1/α competitive buffer policy, which exactly matches the lower bound. We also analyze a RED-like policy and derive its competitive ratio, which is approximately 2α--0.5/α for two values and Θ(log α) for multiple values. In addition we improve the previous known lower and upper bounds of the Fixed Partition and Flexible Partition policies.For the FIFO preemptive model, we improve the general lower bound and show a tight bound for the special case of queue size 2. We prove that the bounded delay model with uniform delay 2 is equivalent to a modified FIFO preemptive model with queue size 2. We then give improved upper and lower bounds on the 2-uniform bounded delay model. We also give lower bound for the 2-variable bounded delay model, which matches the previously known upper bound.