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 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.