We study the problem of centralized allocation of children to public day care centers, illustrated by the case of Denmark. Our framework applies to problems of dynamic matching in which there is entry and exit of agents over time; for example, the school choice problem once student mobility is taken into account. We show that there does not exist any mechanism that is both stable and strategy-proof. We also show that the well-known Top Trading Cycles mechanism is neither Pareto efficient nor strategy-proof. Finally, a mechanism in which parents sequentially choose menus of schools is both strategy-proof and Pareto efficient. (JEL C73, D82, I21)

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In dynamic matching problems, priorities often depend on previous allocations and create opportunities for manipulations that are absent in static problems. In the dynamic school choice problem, students can manipulate the period-by-period deferred acceptance (DA) mechanism. With a commonly used restriction on the schools’ priorities, manipulation vanishes as the number of agents increases, but without it the mechanism can be manipulated, even in large economies. We also check manipulation in large finite economies through a novel computer algorithm, which can check every possible manipulation by examining all the different matchings that a single player can induce. (JEL C78, I21, I28)

The problem of allocating indivisible objects to different agents, where each individual is assigned at most one object, has been widely studied. Pápai (2000) shows that the set of strategy-proof, nonbossy, Pareto optimal and reallocation-proof rules are hierarchical exchange rules -generalizations of Gale's Top Trading Cycles mechanism. We study the centralized allocation that takes place in multiple markets. For example, the assignment of multiple types of indivisible objects; or the assignment of objects in successive periods. We show that the set of strategy-proof, Pareto efficient and nonbossy rules are sequential dictatorships, a special case of Pápai's hierarchical exchange rules.JEL classification: C78, D61, D78, I20.

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