Strategyproofness-Exposing Descriptions of Matching Mechanisms

A menu description exposes strategyproofness by presenting a mechanism to player $i$ in two steps. Step (1) uses others' reports to describe $i$'s menu of potential outcomes. Step (2) uses $i$'s report to select $i$'s favorite outcome from her menu. We provide novel menu descriptions of the Deferred Acceptance (DA) and Top Trading Cycles (TTC) matching mechanisms. For TTC, our description additionally yields a proof of the strategyproofness of TTC's traditional description, in a way that we prove is impossible for DA.