On fairness of multi-center allocation problems

We investigate Ekici (2024b)'s multi-center allocation problems, focusing on fairness in this context. We introduce three fairness notions that respect centers' priorities: internal fairness, external fairness, and procedural fairness. The first notion eliminates envy among agents within the same center, the second prohibits envy across different centers, and the third rules out envy from an ex-ante perspective through agents' trading opportunities. We provide two characterizations of a natural extension of the top-trading-cycles mechanism (TTC) through our fairness notions. Precisely, we show that in the presence of strategy-proofness and pair efficiency, internal fairness and external fairness together characterize TTC (Theorem 1). Also, strategy-proofness combined solely with procedural fairness also characterizes TTC (Theorem 2). Furthermore, by adding internal fairness, we establish our third TTC characterization, by relaxing Ekici's queuewise rationality to another voluntary participation condition, the center lower bound (Theorem 3). Finally, we define a core solution within this model and characterize it through TTC (Theorem 4). Our findings offer practical insights for market designers, particularly in contexts such as international cooperation in medical programs and worker exchange programs.