We consider the problem of locating a single facility on the real line. This facility serves a set of agents, each of whom is located on the line, and incurs a cost equal to his distance from the facility. An agent's location is private information that is known only to him. Agents report their location to a central planner who decides where to locate the facility. The planner's objective is to minimize a "social" cost function that depends on the agent-costs. However, agents might not report truthfully; to address this issue, the planner must restrict himself to strategyproof mechanisms, in which truthful reporting is a dominant strategy for each agent. A mechanism that simply chooses the optimal solution is generally not strategyproof, and so the planner aspires to to use a mechanism that effectively approximates his objective function. This general class of problems was first studied by Procaccia and Tennenholtz and has been the subject of much research since then.In our paper, we study the problem described above with the social cost function being the L p norm of the vector of agent-costs. We show that the median mechanism (which is known to be strategyproof) provides a 2 1− 1 p approximation ratio, and that is the optimal approximation ratio among all deterministic strategyproof mechanisms. For randomized mechanisms, we present two results. First, we present a negative result: we show that for integer ∞ > p > 2, no mechanism-from a rather large class of randomized mechanisms-has an approximation ratio better than that of the median mechanism. This is in contrast to the case of p = 2 and p = ∞ where a randomized mechanism provably helps improve the worst case approximation ratio. Second, for the case of 2 agents, we show that a mechanism called LRM, first designed by Procaccia and Tennenholtz for the special case of L ∞ , provides the optimal approximation ratio among all randomized mechanisms.

In the school choice market, where scarce public school seats are assigned to students, a key operational issue is how to reassign seats that are vacated after an initial round of centralized assignment. Practical solutions to the reassignment problem must be simple to implement, truthful, and efficient while also alleviating costly student movement between schools. We propose and axiomatically justify a class of reassignment mechanisms, the permuted lottery deferred acceptance (PLDA) mechanisms. Our mechanisms generalize the commonly used deferred acceptance (DA) school choice mechanism to a two-round setting and retain its desirable incentive and efficiency properties. School choice systems typically run DA with a lottery number assigned to each student to break ties in school priorities. We show that under natural conditions on demand, the second-round tie-breaking lottery can be correlated arbitrarily with that of the first round without affecting allocative welfare and that reversing the lottery order between rounds minimizes reassignment among all PLDA mechanisms. Empirical investigations based on data from New York City high school admissions support our theoretical findings. This paper was accepted by Gad Allon, operations management.

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