We revisit the problem of designing strategyproof mechanisms for allocating divisible items among two agents who have linear utilities, where payments are disallowed and there is no prior information on the agents' preferences. The objective is to design strategyproof mechanisms which are competitive against the most efficient (but not strategyproof) mechanism. For the case with two items: (1) We provide a set of sufficient conditions for strategyproofness. (2) We use an analytic approach to derive strategyproof mechanisms which are more competitive than all prior strategyproof mechanisms. (3) We improve the linear-program-based proof of Guo and Conitzer to show new upper bounds on competitive ratios. (4) We provide the first "mathematical" upper bound proof. For the cases with any number of items: (1) We build on the Partial Allocation mechanisms introduced by to design a strategyproof mechanism which is 0.67776-competitive, breaking the 2 3 barrier. (2) We propose a new subclass of strategyproof mechanisms called Dynamical-Increasing-Price mechanisms, where each agent purchases the items using virtual money, and the prices of the items depend on other agents' preferences.
Cole, Gkatzelis and Goel @cite_5 proposed another sub-class of SP mechanisms (for any number of agents) called (PA) mechanisms, which are not swap-dictatorial. They showed in another work @cite_6 that a variant of PA mechanism is @math -competitive for two agents and any number of items.
@cite_5 @cite_6
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In this paper we study the problem of allocating divisible items to agents without payments. We assume no prior knowledge about the agents. The utility of an agent is additive. The social welfare of a mechanism is defined as the overall utility of all agents. This model is first defined by Guo and Conitzer [7]. Here we are interested in strategy-proof mechanisms that have a good competitive ratio, that is, those that are able to achieve social welfare close to the maximal social welfare in all cases. First, for the setting of n agents and m items, we prove that there is no (1 m+e)-competitive strategy-proof mechanism, for any e>0. And, no mechanism can achieve a competitive ratio better than @math , when @math . Next we study the setting of two agents and m items, which is also the focus of [7]. We prove that the competitive ratio of any swap-dictatorial mechanism is no greater than @math . Then we give a characterization result: for the case of 2 items, if the mechanism is strategy-proof, symmetric and second order continuously differentiable, then it is always swap-dictatorial. In the end we consider a setting where an agent's valuation of each item is bounded by C m, where C is an arbitrary constant. We show a mechanism that is (1 2+e(C))-competitive, where e(C)>0. The authors show how to design truthful (dominant strategy) mechanisms for several combinatorial problems where each agent's secret data is naturally expressed by a single positive real number. The goal of the mechanisms we consider is to allocate loads placed on the agents, and an agent's secret data is the cost she incurs per unit load. We give an exact characterization for the algorithms that can be used to design truthful mechanisms for such load balancing problems using appropriate side payments. We use our characterization to design polynomial time truthful mechanisms for several problems in combinatorial optimization to which the celebrated VCG mechanism does not apply. For scheduling related parallel machines (Q spl par C sub max ), we give a 3-approximation mechanism based on randomized rounding of the optimal fractional solution. This problem is NP-complete, and the standard approximation algorithms (greedy load-balancing or the PTAS) cannot be used in truthful mechanisms. We show our mechanism to be frugal, in that the total payment needed is only a logarithmic factor more than the actual costs incurred by the machines, unless one machine dominates the total processing power. We also give truthful mechanisms for maximum flow, Q spl par spl Sigma C sub j (scheduling related machines to minimize the sum of completion times), optimizing an affine function over a fixed set, and special cases of uncapacitated facility location. In addition, for Q spl par spl Sigma w sub j C sub j (minimizing the weighted sum of completion times), we prove a lower bound of 2 spl radic 3 for the best approximation ratio achievable by truthful mechanism. We study strategyproof (SP) mechanisms for the location of a facility on a discrete graph. We give a full characterization of SP mechanisms on lines and on sufficiently large cycles. Interestingly, the characterization deviates from the one given by Schummer and Vohra (2004) for the continuous case. In particular, it is shown that an SP mechanism on a cycle is close to dictatorial, but all agents can affect the outcome, in contrast to the continuous case. Our characterization is also used to derive a lower bound on the approximation ratio with respect to the social cost that can be achieved by an SP mechanism on certain graphs. Finally, we show how the representation of such graphs as subsets of the binary cube reveals common properties of SP mechanisms and enables one to extend the lower bound to related domains. We consider the problem of locating a facility on a network represented by a graph. A set of strategic agents have different ideal locations for the facility; the cost of an agent is the distance between its ideal location and the facility. A mechanism maps the locations reported by the agents to the location of the facility. We wish to design mechanisms that are strategyproof (SP) in the sense that agents can never benefit by lying and, at the same time, provide a small approximation ratio with respect to the minimax measure. We design a novel “hybrid” strategyproof randomized mechanism that provides a tight approximation ratio of 3 2 when the network is a circle (known as a ring in the case of computer networks). Furthermore, we show that no randomized SP mechanism can provide an approximation ratio better than 2 - o (1), even when the network is a tree, thereby matching a trivial upper bound of two.