The US National Residency Matching Program (NRMP) is a computerized, national system for matching residency applicants to programs. Similar systems exist in Canada, Australia, and New Zealand, and the need for such a program will probably make itself felt in the European Union soon. NRMP is an important laboratory for the ethical challenges that computerized matching programs create, especially its current prohibition of making commitments by both applicants and residency programs. Nonetheless, it would be denial to suppose that commitments are not being made and that there is no dishonesty when commitments are made. We analyze the current NRMP no-commitment policy by reference to Plato's retelling of the myth of the Ring of Gyges, which makes its wearer physically invisible and therefore ethically unaccountable. Because applicants and programs complete their preference-lists in secret, the matching process inadvertently slips a Ring of Gyges on the fingers of both parties. As a result, making false commitments promotes the rational self-interest of both parties. To strengthen the professional integrity of the matching process, we argue that NRMP should abandon its no-commitment policy and adopt a new paradigm: sanctioning honest, non-contingent, accountable, one-party, and documented commitments. Computerized residency matching systems in other countries should consider the NRMP a laboratory and develop policies about making commitments that promote professionalism in the residency application process.