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The one-sided matching problem is concerned with the allocation of indivisible goods to self-interested agents with privately known preferences. Monetary transfers are not permitted, which makes this problem different from auctions and other settings with transferable utility. In practice, such problems often arise in situations that are of great importance to peoples' lives. For example, students must be matched to schools, teachers to training programs, or houses to tenants. While strategyproofness is certainly a desirable design desideratum, it is also a severe restriction when other properties are important, such as efficiency or fairness. We study ordinal mechanisms for this problem, where agents have vNM utility functions over the objects. The paper makes two main contributions: we 1. characterize strategyproof one-sided matching mechanisms by three intuitive axioms, and we 2. propose the partial strategyproofness concept, a relaxation of strategyproofness, which bridges the long-standing gap between full and weak strategyproofness. CHARACTERIZATION OF STRATEGYPROOF MECHANISMSOur first contribution is to show that strategyproof mechanisms are characterized by three intuitive axioms. To understand the axioms, suppose an agent is considering whether to report truthfully or swap two adjacent objects, e.g., a b to b a:(1) A mechanism is swap monotonic if upon such a swap, the reporting agent's allocation either does not change at all, or the allocation for b strictly increases and the allocation for a strictly decreases.(2) A mechanism is upper invariant if the allocation does not change for any object that the agent strictly prefers to a, i.e., any object in the upper contour set of a.(3) The mechanism is lower invariant if the allocation does not change for any object that the agent likes strictly less than b, i.e., any object in the lower contour set of b.We show that a mechanism is strategyproof if and only if it is swap monotonic, upper invariant, and lower invariant. Thus, strategyproofness requires that the mechanism only affects the allocation of the objects that are swapped (if any), and the direction of this change must be consistent with the agent's reported preferences.

We study the trade-offs between strategyproofness and other desiderata, such as efficiency or fairness, that often arise in the design of random ordinal mechanisms. We use ε-approximate strategyproofness to define manipulability, a measure to quantify the incentive properties of non-strategyproof mechanisms, and we introduce the deficit, a measure to quantify the performance of mechanisms with respect to another desideratum. When this desideratum is incompatible with strategyproofness, mechanisms that trade off manipulability and deficit optimally form the Pareto frontier. Our main contribution is a structural characterization of this Pareto frontier, and we present algorithms that exploit this structure to compute it. To illustrate its shape, we apply our results for two different desiderata, namely Plurality and Veto scoring, in settings with 3 alternatives and up to 18 agents. comments on this work. Furthermore, we are thankful for the feedback we received from participants at COST COM-SOC Meeting 2016 (Istanbul, Turkey) and multiple anonymous referees at EC'15 and EC'16. Any errors remain our own. A 1-page abstract based on this paper has been published in the conference proceedings of EC'16 (Mennle and Seuken, 2016).

We present partial strategyproofness, a new, relaxed notion of strategyproofness for studying the incentive properties of non-strategyproof assignment mechanisms. Informally, a mechanism is partially strategyproof if it makes truthful reporting a dominant strategy for those agents whose preference intensities differ sufficiently between any two objects. We demonstrate that partial strategyproofness is axiomatically motivated and that it provides a unified perspective on incentives in the assignment domain. We then apply it to derive novel insights about the incentive properties of the Probabilistic Serial mechanism and different variants of the Boston mechanism.