We consider the stability of matchings when individuals strategically submit preference information to a publicly known algorithm. Most pure Nash equilibria of the ensuing game yield a matching that is unstable with respect to the individuals sincere preferences. We introduce a well-supported minimal dishonesty constraint, and obtain conditions under which every pure Nash equilibrium yields a matching that is stable with respect to the sincere preferences. The conditions on the matching algorithm are to be either fully-randomized, or monotonic and independent of non-spouses (INS), an IIA-like property. These conditions are significant because they support the use of algorithms other than the Gale-Shapley (man-optimal) algorithm for kidney exchange and other applications. We prove that the Gale-Shapley algorithm always yields the woman-optimal matching when individuals are minimally dishonest. However, we give a negative answer to one of Gusfield and Irvings open questions: there is no monotonic INS or fully-randomized stable matching algorithm that is certain to yield the egalitarian-optimal matching when individuals are strategic and minimally dishonest. Finally, we show that these results extend to the student placement problem, where women are polyandrous but must be honest but do not extend to the admissions problem, where women are both polyandrous and strategic.
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