We study the structure of the set of priority-neutral matchings. These matchings, introduced by [Ren22], generalize stable matchings by allowing for priority violations in a principled way that enables Paretoimprovements to stable matchings. Known results show that the set of priority-neutral matchings is a lattice, suggesting that these matchings may enjoy the same tractable theoretical structure as stable matchings. In this paper, we characterize priority-neutral matching lattices, and show that their structure is considerably more intricate than that of stable matching lattices. To begin, we show priority-neutral lattices are not distributive, an important property that characterizes stable lattices and is satisfied by many other lattice structures considered in matching theory and algorithm design. Then, in our main result, we show that priority-neutral lattices are in fact characterized by a more-involved property which we term being a “movement lattice,” which allows for significant departures from the order theoretic properties of distributive (and hence stable) lattices. While our results show that priority-neutrality is more intricate than stability, they also establish tractable properties. Indeed, as a corollary of our main result, we obtain the first known polynomialtime algorithm for checking whether a given matching is priority-neutral.

We study the communication complexity of truthful combinatorial auctions, and in particular the case where valuations are either subadditive or single-minded, which we denote with SubAddUSingleM. We show that for three bidders with valuations in SubAddUSingleM, any deterministic truthful mechanism that achieves at least a 0.366-approximation requires$\exp(m)$communication. In contrast, a natural extension of [Fei09] yields a non-truthful$\text{poly}(m)-\mathbf{communication}$protocol that achieves a$\frac{1}{2}-\mathbf{approximation}$, demonstrating a gap between the power of truthful mechanisms and non-truthful protocols for this problem. Our approach follows the taxation complexity framework laid out in [Dob16b], but applies this framework in a setting not encompassed by the techniques used in past work. In particular, the only successful prior application of this framework uses a reduction to simultaneous protocols which only applies for two bidders [AKSW20], whereas our three-player lower bounds are stronger than what can possibly arise from a two-player construction (since a trivial truthful auction guarantees a$\frac{1}{2}- \mathbf{approximation}$for two players).

Transaction Fee Mechanism Design---a rapidly-evolving research agenda initiated by Roughgarden [2021]---studies auctions run by untrusted miners for transaction inclusion in a blockchain. Under previously-considered desiderata, an auction is considered 'good' if, informally-speaking, each party (i.e., the miner, the users, and coalitions of both miners and users) has no incentive to deviate from the fixed and pre-determined protocol. In other words, previous works posit that a 'good' auction should be 'simple for users', 'simple for miners', and 'resistant to collusion'.

We conduct an incentivized lab experiment to test participants' ability to understand the DA matching mechanism and the strategyproofness property, conveyed in different ways. We find that while many participants can (using a novel GUI) learn DA's mechanics and calculate its outcomes, such understanding does not imply understanding of strategyproofness (as measured by specially designed tests). However, a novel menu description of strategyproofness conveys this property significantly better than other treatments. While behavioral effects are small on average, participants with levels of strategyproofness understanding above a certain threshold play the classical dominant strategy at very high rates.

We study various novel complexity measures for two-sided matching mechanisms, applied to the two canonical strategyproof matching mechanisms, Deferred Acceptance (DA) and Top Trading Cycles (TTC). Our metrics are designed to capture the complexity of various structural (rather than computational) concerns, in particular ones of recent interest within economics. We consider a unified, flexible approach to formalizing our questions: Define a protocol or data structure performing some task, and bound the number of bits that it requires. Our main results apply this approach to four questions of general interest; for mechanisms matching applicants to institutions, our questions are: (1) How can one applicant affect the outcome matching? (2) How can one applicant affect another applicant's set of options? (3) How can the outcome matching be represented / communicated? (4) How can the outcome matching be verified? Holistically, our results show that TTC is more complex than DA, formalizing previous intuitions that DA has a simpler structure than TTC. For question (2), our result gives a new combinatorial characterization of which institutions are removed from each applicant's set of options when a new applicant is added in DA; this characterization may be of independent interest. For question (3), our result gives new tight lower bounds proving that the relationship between the matching and the priorities is more complex in TTC than in DA. We nonetheless showcase that this higher complexity of TTC is nuanced: By constructing new tight lower-bound instances and new verification protocols, we prove that DA and TTC are comparable in complexity under questions (1) and (4). This more precisely delineates the ways in which TTC is more complex than DA, and emphasizes that diverse considerations must factor into gauging the complexity of matching mechanisms.

A menu description presents a mechanism to player i in two steps. Step (1) uses the reports of other players to describe i's menu: the set of i's potential outcomes. Step (2) uses i's report to select i's favorite outcome from her menu. Can menu descriptions better expose strategyproofness, without sacrificing simplicity? We propose a new, simple menu description of Deferred Acceptance. We prove that---in contrast with other common matching mechanisms---this menu description must differ substantially from the corresponding traditional description. We demonstrate, with a lab experiment on two elementary mechanisms, the promise and challenges of menu descriptions.

We study the stable marriage problem in two-sided markets with randomly generated preferences. We consider agents on each side divided into a constant number of "soft tiers", which intuitively indicate the quality of the agent. Specifically, every agent within a tier has the same public score, and agents on each side have preferences independently generated proportionally to the public scores of the other side. We compute the expected average rank which agents in each tier have for their partners in the men-optimal stable matching, and prove concentration results for the average rank in asymptotically large markets. Furthermore, we show that despite having a significant effect on ranks, public scores do not strongly influence the probability of an agent matching to a given tier of the other side. This generalizes results of [Pittel 1989] which correspond to uniform preferences. The results quantitatively demonstrate the effect of competition due to the heterogeneous attractiveness of agents in the market, and we give the first explicit calculations of rank beyond uniform markets.

We study the strategic simplicity of stable matching mechanisms where one side has fixed preferences, termed priorities. Specifically, we ask which priorities are such that the strategyproofness of deferred acceptance (DA) can be recognized by agents unable to perform contingency reasoning, that is, when does DA have an obviously strategyproof (OSP) implementation (Li, 2017)? We answer this question by completely characterizing those priorities for which DA is OSP implementable. This solves an open problem of Ashlagi and Gonczarowski, 2018. We find that when DA is OSP implementable, priorities are either acyclic (Ergin, 2002), a restrictive condition which allows priorities to only differ on only two agents at a time, or contain an extremely limited cyclic pattern where all priority lists are identical except for exactly two. We conclude that, for stable matching mechanisms, the tension between understandability (in the sense of OSP) and expressiveness of priorities is very high.

We consider the problem of implementing a fixed social choice function between multiple players (which takes as input a type ti from each player i and outputs an outcome f(t1,…, tn)), in which each player must be incentivized to follow the protocol. In particular, we study the communication requirements of a protocol which: (a) implements f, (b) implements f and computes payments that make it ex-post incentive compatible (EPIC) to follow the protocol, and (c) implements f and computes payments in a way that makes it dominant-strategy incentive compatible (DSIC) to follow the protocol.

State-of-the-art posted-price mechanisms for submodular bidders with m items achieve approximation guarantees of O((log log m)^3) [Sepehr Assadi and Sahil Singla, 2019]. Their truthfulness, however, requires bidders to compute an NP-hard demand-query. Some computational complexity of this form is unavoidable, as it is NP-hard for truthful mechanisms to guarantee even an m^(1/2-ε)-approximation for any ε > 0 [Shahar Dobzinski and Jan Vondrák, 2016]. Together, these establish a stark distinction between computationally-efficient and communication-efficient truthful mechanisms. We show that this distinction disappears with a mild relaxation of truthfulness, which we term implementation in advised strategies. Specifically, advice maps a tentative strategy either to that same strategy itself, or one that dominates it. We say that a player follows advice as long as they never play actions which are dominated by advice. A poly-time mechanism guarantees an α-approximation in implementation in advised strategies if there exists advice (which runs in poly-time) for each player such that an α-approximation is achieved whenever all players follow advice. Using an appropriate bicriterion notion of approximate demand queries (which can be computed in poly-time), we establish that (a slight modification of) the [Sepehr Assadi and Sahil Singla, 2019] mechanism achieves the same O((log log m)^3)-approximation in implementation in advised strategies.