We study a many-to-one assignment problem, where a set of applicants need to be assigned to a set of schools with limited seats. We formulate distributional objectives by a set of standard axioms. Axiom 1 has two parts. First, the number of applicants of each type at each school should be less than the corresponding quota. Second, whenever the number of applicants of a given type assigned to a school is less than the corresponding reserve, then it should be that no other applicant of that type prefers the school to their own assignment. Axiom 2 requires that whenever a school has available seats, it should be that there is no applicant that prefers that school to their own assignment. Finally, we say that an applicant a violates the priority of another applicant b at school s, if b prefers s to their own assignment, and b has a higher priority at that school than a. In this case, the triplet (a, b, s) is called a priority violation instance. Axiom 3 requires that there are no priority violation instances including two applicants of the same type.

In centralized market mechanisms individuals may not fully observe other individuals' type reports. Hence, the mechanism designer may deviate from the promised mechanism without the individuals being able to detect these deviations. In this paper, we develop a theory of auditability for allocation and social choice problems. Namely, we measure a mechanism's auditability by the smallest number of individuals that can jointly detect any deviation. Our theory reveals stark contrasts between prominent mechanisms' auditability properties in various applications. For priority-based allocation problems, we find that the Immediate Acceptance mechanism is maximally auditable, in a sense that any deviation can always be detected by just two individuals. On the other extreme, the Deferred Acceptance mechanism is minimally auditable, in a sense that some deviations may go undetected unless the type reports of everyone are known. For a class of mechanisms that can be implemented as Deferred Acceptance in systematically modified problems, we establish a relation between a mechanism's auditability and the uniqueness of stable outcomes in the modified problems. For single-unit auction problems, we show that under the first-price and the all-pay auction mechanisms just two individuals can detect any deviation, whereas the second-price auction mechanism requires full information for detecting an adversarial deviation. For voting problems with a binary outcome, we characterize the dictatorial rule as the unique voting mechanism under which any deviation can be detected by a single individual. Moreover, we characterize the majority voting rule as the unique most auditable anonymous voting mechanism. Finally, for the choice with affirmative action setting, we compare the auditability of two prominent reserves mechanisms, establishing the superiority of one mechanism over the other.