Some exchange programs require balance--such as exchanges of students among colleges, e.g. for study abroad. Here's a paper addressing that in the JPE, with some impossibility results and a constrained optimality approach:
Two-Sided Matching via Balanced Exchange
Umut Mert Dur, North Carolina State University
and
M. Utku Ünver, Boston College and Deakin University
Journal of Political Economy 127, no. 3 (June 2019): 1156-1177.
Abstract We introduce a new matching model to mimic two-sided exchange programs such as tuition and worker exchanges, in which export-import balances are required for longevity of programs. These exchanges use decentralized markets, making it difficult to achieve this goal. We introduce the two-sided top trading cycles, the unique mechanism that is balanced-efficient, worker-strategy-proof, acceptable, individually rational, and respecting priority bylaws regarding worker eligibility. Moreover, it encourages exchange, because full participation induces a dominant-strategy equilibrium for firms. We extend it to dynamic settings permitting tolerable yearly imbalances and demonstrate that its regular and tolerable versions perform considerably better than models of current practice.
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Here's an earlier post about an exchange program of the kind addressed in the paper:
Two-Sided Matching via Balanced Exchange
Umut Mert Dur, North Carolina State University
and
M. Utku Ünver, Boston College and Deakin University
Journal of Political Economy 127, no. 3 (June 2019): 1156-1177.
Abstract We introduce a new matching model to mimic two-sided exchange programs such as tuition and worker exchanges, in which export-import balances are required for longevity of programs. These exchanges use decentralized markets, making it difficult to achieve this goal. We introduce the two-sided top trading cycles, the unique mechanism that is balanced-efficient, worker-strategy-proof, acceptable, individually rational, and respecting priority bylaws regarding worker eligibility. Moreover, it encourages exchange, because full participation induces a dominant-strategy equilibrium for firms. We extend it to dynamic settings permitting tolerable yearly imbalances and demonstrate that its regular and tolerable versions perform considerably better than models of current practice.
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Here's an earlier post about an exchange program of the kind addressed in the paper: