Optimal allocation without money: an engineering approach

We study the optimal allocation of heterogeneous services without using monetary transfers. Agents have private, multi-dimensional utilities over the services, and a social planner has arbitrary priors on the utilities, which may depend on the agents' observable characteristics. The social planner's goal is to maximize a public objective, which may be complex, taking into account diverse considerations such as social welfare, equity, and system costs. Potential applications include the allocation of seats to public schools, spaces in college dorms or courses, and spots in subsidized housing. We propose an "engineering" approach to such problems, in which we simplify the analysis by considering a large-market model with a continuum of agents, and show using real data that a mechanism defined based on this approximation still yields good results in the finite market. In the large-market model, there are finitely many "agent types," and a continuum of agents of each type. (Here, "type" corresponds to an agent's observable information, which is distinct from the private utilities.) The large market assumption implies that a mechanism can be decomposed into a collection of allocation rules, one for each type. We show that any allocation rule that satisfies incentive compatibility and Pareto optimality within type is representable as "Competitive Equilibrium with Equal Incomes" (CEEI). In other words, for each type of agents and for each service, there exists a "virtual price" for a unit of probability of getting that service, and the allocation is induced by giving each agent one unit of "virtual money' and having them purchase their utility-maximizing probabilities to services. In addition, if the mechanism is restricted to being 'ordinal,' which means that it only uses preference rankings but not preference intensities, then an allocation rule that satisfies incentive compatibility and ordinal efficiency within type is 'lottery-plus-cutoff': each agent receives a uniformly random lottery number between zero and one, and for each service and each type, there is a 'lottery cutoff'; an agent is 'admitted' to a service if her lottery number is below the cutoff; each agent is allocated her most preferred service for which she is admitted. Such characterization results reduce the search of the optimal mechanism to a well-defined optimization over the prices and cutoffs for each type. We show how this large market approximation can be applied to an empirically relevant finite setting and yield good results. We consider the allocation of seats in public schools in Boston, which was a real problem faced by a city committee in the 2012-2013 Boston school assignment reform. Students are classified into 868 types by home location and there is a utility model for each type that comes from fitting a multinomial logit discrete choice model on previous years' data. There are 77 elementary schools with given capacities. Depending on their home location, a student is given a menu of options and can submit a preference ranking over the schools in the menu. Given preference submissions, a centralized algorithm computes the assignment using a system of priorities, possibly randomizing to break ties. The goal is to optimize the menus and priorities to maximize a weighted combination of social welfare and max-min welfare, while staying within a budget for expected amount of busing needed. Although this is a finite market problem, we can define a large market approximation. Using the characterization results, we show that the optimal mechanism can be encoded as an exponential-sized linear program, which due to the logit utility structure can be efficiently solved using duality theory. Using the cutoffs that arise from this optimization, we define a feasible finite market mechanism that is based on the Deferred Acceptance algorithm. We evaluate the resultant mechanism by simulating in the finite market setting with real data, and show that compared to the mechanism chosen by the city, this new mechanism significantly improves simultaneously in social welfare, max-min welfare and predictability, while staying within the same expected busing budget. In sum, we incorporate prior information in the allocation problem without monetary transfers, and exhibit computational results in empirically relevant special cases.