While the seminal paper on assignment mechanisms by Hylland and Zeckhauser (1979) proposed a mechanism that elicits agents’ cardinal utilities, this approach has proven problematic because it is difficult if not impossible to elicit cardinal utilities in settings without money. For this reason, recent work has focused on ordinal mechanisms where
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2.2 Related Work agents submit preference orders over objects. In fact, it has been shown that mechanisms whose outcomes have to be independent of the agents’ levels of wealth are bound to be ordinal (Huesmann and Wambach, 2015; Ehlers et al., 2015). Throughout this paper, we only consider ordinal mechanisms.
For the case of deterministic assignment mechanisms, strategyproofness has been studied extensively. P´apai (2000) showed that the only group-strategyproof, ex-post efficient, reallocation-proof mechanisms are hierarchical exchanges. Characterizations of strategyproof, efficient, and reallocation-consistent (Ehlers and Klaus,2006) or consis- tent (Ehlers and Klaus, 2007) mechanisms are also available. The only deterministic, strategyproof, ex-post efficient, non-bossy, and neutral mechanisms are known to be serial dictatorships (Hatfield, 2009). Furthermore, Pycia and ¨Unver (2014) showed that all group-strategyproof, ex-post efficient mechanisms are trading cycles mechanisms.
Barbera, Berga and Moreno (2012) gave a decomposition of strategyproofness that is similar in spirit to ours but restricted to deterministic social choice domains.
For random social choice rules, Gibbard (1977) gave a decomposition of strategyproof- ness into the two properties localized and non-perverse. This (as well as any other) decomposition of strategyproofness is by definition equivalent to our decomposition. Here our contribution lies in the definition of simple and intuitive axioms that make the con- ditions accessible and straightforward to interpret. For random assignment mechanisms,
Abdulkadiro˘glu and S¨onmez (1998) showed that Random Serial Dictatorship (RSD) is equivalent to the Core from Random Endowments mechanism for house allocation. Bade
(2014) extended their result by showing that taking any ex-post efficient, strategyproof, non-bossy, deterministic mechanism and assigning agents to roles in the mechanism uniformly at random is equivalent to RSD. However, it is still an open conjecture whether RSD is the unique mechanism that is strategyproof, ex-post efficient, and anonymous (Lee and Sethuraman, 2011; Bade, 2014).
Besides the baseline requirement of ex-post efficiency, the research community has also introduced more demanding efficiency concepts, such as ordinal efficiency, which is achieved by the Probabilistic Serial (PS) mechanism (Bogomolnaia and Moulin, 2001). The PS mechanism has received considerable attention from researchers: Hashimoto et al.
(2014) showed that PS with uniform eating speeds is in fact the unique mechanism that is ordinally fair and non-wasteful. Bogomolnaia and Moulin (2001) had already shown that PS is not strategyproof but Ekici and Kesten (2012) found that its Nash equilibria can lead to ordinally dominated outcomes. Incentive concerns for PS may be severe for small settings but they get less problematic for larger settings: Kojima and Manea
(2010) showed that for a fixed number of object types and a fixed agent PS makes it a dominant strategy for that agent to be truthful if the number of copies of each object is sufficiently large.
While ex-post efficiency and ordinal efficiency are the most well-studied efficiency concepts for assignment mechanisms, some mechanisms used in practice aim to achieve
rank efficiency which is a further refinement of ordinal efficiency (Featherstone,2011). However, no rank efficient mechanism can be strategyproof in general. Other popular mechanisms, like the Boston Mechanism (Ergin and S¨onmez, 2006; Miralles, 2008), are highly manipulable but nevertheless in frequent use. Budish and Cantillon (2012) found practical evidence from combinatorial course allocation which suggests that using a non-strategyproof mechanism may lead to higher social welfare than using an ex-post efficient and strategyproof mechanism, such as RSD. The fact that strategyproofness is in conflict with many other design objectives challenges whether it should be taken as an indispensable requirement in mechanism design.
Given that strategyproofness is such a strong restriction, many researchers have tried to relax it. Bogomolnaia and Moulin (2001) used weak SD-strategyproofness to describe the incentive properties under PS andBalbuzanov(2015) showed that PS in fact satisfies the more demanding convex strategyproofness. Carroll (2013) adapted approximate
strategyproofness for bounded utilities to quantify agents’ incentives to manipulate in the
voting domain. Azevedo and Budish (2015) proposed a desideratum called strategyproof
in the large (SP-L) which formalizes the intuition that as the number of agents in the
market gets large the incentives for an individual agent to misreport its preference order should vanish in the limit. Finally, Cho (2012) considered strategyproofness for agents with lexicographic preferences (DL-strategyproofness). We show that partial strategyproofness unifies these relaxations of strategyproofness: on the one hand, many non-strategyproof mechanisms that are generally viewed as “having better incentive properties,” because they satisfy these various notions of strategyproofness, turn out to satisfy partial strategyproofness as well, such as Probabilistic Serial, a variant of the Boston Mechanism, and newly defined hybrid mechanisms. On the other hand, partial strategyproofness implies the other notions. Pathak and S¨onmez (2013) introduced a general method to compare different mechanisms by their vulnerability to manipulation. The degree of strategyproofness measure we propose in this paper is consistent with (but not equivalent to) this method. However, our concept has two advantages: it is parametric and it is computable. We discuss the connection in more detail in Section
2.3 Model