Asymptotic Welfare Performance of Boston Assignment Algorithms

We make a detailed analysis in a special case of the Boston algorithm, which is widely used around the world to assign students to schools. We compute the limiting distribution in large random markets of both the utilitarian welfare and the order bias, a recently introduced average-case fairness measure. Our results show that the differences in utilitarian welfare between the Boston algorithms and the serial dictatorship (SD) algorithm are small and positive, whereas the differences in terms of order bias are large and positive. The naive implementation of the Boston algorithm beats its adaptive implementation on both utilitarian welfare and order bias, and both apparently beat SD on both criteria. In order to establish our results, we derive several basic results on the time evolution of the assignments made by the algorithms, which we expect to be useful for other applications. For example, we compute limiting distributions as a function of [Formula: see text] of the exit time and preference rank obtained for an arbitrary agent whose initial relative position in the tiebreak order is θ.