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.
@cite_13 showed a number of upper bound results on the competitiveness of SP mechanisms, when the numbers of agents and or items increase. In particular, they showed that no swap-dictatorial mechanism can be better than @math -competitive for @math items. In addition, they proved the following characterization result: in the case with two items, if a mechanism @math is symmetric and second order continuously differentiable, then @math is SP if and only if @math is swap-dictatorial.
@cite_13
176406899 2106615115 2063004177 2107435
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. 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 study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomized mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomized mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of eovere-1. The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of 18 over 13 on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. In the case that the mechanism must additionally be non-bossy, an improved lower bound of eovere-1 holds. This lower bound is tight given that RSDM for strict preference lists is non-bossy. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomized strategy of the administrator who interviews the applicants. Consider the problem of allocating multiple divisible goods to two agents in a strategy-proof fashion without the use of payments or priors. Previous work has aimed at implementing allocations that are competitive with respect to an appropriately defined measure of social welfare. These results have mostly been negative, proving that no dictatorial mechanism can achieve an approximation factor better than 0.5, and leaving open the question of whether there exists a non-dictatorial mechanism that outperforms this bound. We provide a positive answer to this question by presenting an interesting non-dictatorial mechanism that achieves an approximation factor of 2 3 for this measure of social welfare. In proving this bound we also touch on the issue of fairness: we show that the proportionally fair solution, a well known fairness concept for money-free settings, is highly competitive with respect to social welfare. We then show how to use the proportionally fair solution to design our non-dictatorial strategy-proof mechanism.