Optimal Allocation of Public Services Without Monetary Transfers or Costly Signals

We study the social planner’s problem of designing a mechanism to optimally allocate a set of public services to a set of heterogeneous agents with private utilities, without the ability to differentiate agents by charging prices or requiring costly effort. This is motivated by the 2012-2013 Boston school choice reform, in which social planners had to design a school choice lottery to allocate public school seats among families from various neighborhoods, in order to balance efficiency, equity, and busing costs. We consider two types of mechanisms: cardinal mechanisms, which can use any information, and ordinal mechanisms, which can only use agents’ rankings of preferences but not preference intensities. We show that under assumptions of “soft capacity limits” and “Pareto optimality of interim allocation rules,” any valid interim cardinal allocation rule is “type-specific-pricing,” in which agents are given equal budgets of “virtual money” and buy the optimal probabilistic bundle of services given type-specific prices for each service. Under similar assumptions, any valid ordinal allocation rule is “lottery-plus-cutoff,” in which agents are given i.i.d. lottery numbers and services post typespecific lottery-cutoffs; agents get their most preferred service for which they do not exceed the cutoff. Given additional assumptions on the public objective function and allowing for linear budget constraints, we present an algorithm to efficiently compute the optimal ordinal mechanism. We apply this to real data from Boston and for each of the main plans proposed during the 2012-2013 reform, we compute a corresponding “optimal” plan that uses the same transportation budget but optimizes for efficiency and equity. We compare the plans and discuss potential policy insights.