  Lying in order to manipulate the Gale-Shapley matching algorithm has been
studied by Dubins and Friedman and by Gale and Sotomayor and was shown to be
generally more appealing to the proposed-to side (denoted as the women in Gale
and Shapley's original paper) than to the proposing side (denoted as men
there). It can also be shown that in the case of lying women, for every woman
who is better-off due to lying, there exists a man who is worse-off.
  In this paper, we show that an even stronger dichotomy between the goals of
the sexes holds, namely, if no woman is worse-off then no man is better-off,
while a form of sisterhood between the lying and the "innocent" women also
holds, namely, if none of the former are worse-off, then neither is any of the
latter.
  This paper is based upon an undergraduate ("Amirim") thesis of the first
author.
