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 from economics. We consider a canonical, 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 results apply this approach to four questions of general interest; for matching applicants to institutions, we ask: (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? We prove that DA and TTC are comparable in complexity under questions (1) and (4), giving new tight lower-bound constructions and new verification protocols. Under questions (2) and (3), we prove that TTC is more complex than DA. For question (2), we prove this by giving a new 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 lower bounds proving the tightness of existing constructions for TTC. This shows that the relationship between the matching and the priorities is more complex in TTC than in DA, formalizing previous intuitions from the economics literature. Together, our results complement recent work that models the complexity of observing strategyproofness and shows that DA is more complex than TTC. This emphasizes that diverse considerations must factor into gauging the complexity of matching mechanisms.

翻译：我们研究了两类经典防策略匹配机制——延迟接受（DA）与顶级交易循环（TTC）——的各种新颖复杂度度量指标。这些指标旨在捕捉结构（而非计算）层面的复杂性特征，尤其关注近期经济学研究中的热点问题。我们采用一种规范且灵活的方法来形式化研究问题：定义执行特定任务的协议或数据结构，并约束其所需的比特数。我们的成果将这一方法应用于四个通用性问题：针对申请者与机构的匹配场景，我们提出：（1）单个申请者如何影响最终匹配结果？（2）单个申请者如何影响其他申请者的可选方案集？（3）匹配结果如何表示/传达？（4）匹配结果如何验证？我们证明，在问题（1）和（4）下，DA和TTC具有可比性的复杂度，并给出了新的严格下界构造与验证协议。而在问题（2）和（3）下，我们证明TTC比DA更复杂。针对问题（2），我们通过给出DA机制中新增申请者导致各申请者可选机构集删减的全新表征来证明这一结论；该表征可能具有独立研究价值。针对问题（3），我们的结果提供了下界证明，验证了TTC现有构造的紧致性。这表明TTC中匹配结果与优先级之间的关联比DA更复杂，形式化了经济学文献中的既有直觉。综合来看，我们的研究补充了近期关于策略可观察性复杂度建模的工作——该工作表明DA比TTC更复杂。这凸显了评估匹配机制复杂度时需综合考量多元因素。