Sequential composition is a natural way to build an aggregate choice function from component choice functions. As the name suggests, the components are ordered and choose sequentially. The sets that subsequent components select from are constrained by the choices made by earlier choice functions. The specific constraints affect whether properties like path independence are preserved. For several domains of inputs, we characterize the constraints that ensure such preservation.

For object reallocation problems, Bade (2019) defines a new rule, the “crawler”, and shows that on the domain of single-peaked preferences, this rule satisfies efficiency, the endowments lower bounds, and obvious strategy-proofness. We generalize the crawler, obtaining a family of rules that we call “generalized crawlers”, and show that a rule satisfies efficiency, the endowments lower bounds, and obvious strategy-proofness if and only if it is a generalized crawler. We also provide a structural analysis of the family of generalized crawlers.

n the Assignment Game, introduced by Shapley and Shubik (1971), most solution concepts yield a multiplicity of solutions. We study the Assignment Game in a Bayesian environment where neither buyers nor sellers know the valuation of other players, and derive conditions on the distribution of valuations to guarantee the uniqueness of equilibrium. We also provide a closed-form solution when valuations follow an exponential distribution. Finally, we observe that the Intermediate Value Theorem is pervasive in auction settings.

We propose three basic welfare axioms for school choice mechanisms with an affirmative action policy: non-wastefulness, respecting the affirmative action policy, and minimal responsiveness, and show that none of the previously proposed mechanisms satisfy all three welfare axioms. Then we introduce a new mechanism which satisfies the three welfare axioms. This mechanism issues immediate acceptances to minority students for minority reserve seats and otherwise it is based on deferred acceptance. We also analyze the fairness and incentive properties of this newly proposed affirmative action mechanism.