In this paper, we consider permutation manipulations by any subset of women in the Gale-Shapley algorithm. This paper is motivated by the college admissions process in China. Our results also answer Gusfield and Irvings open problem on what can be achieved by permutation manipulations. We present an efficient algorithm to find a strategy profile such that the induced matching is stable and Pareto-optimal while the strategy profile itself is inconspicuous. Surprisingly, we show that such a strategy profile actually forms a Nash equilibrium of the manipulation game. We also show that a strong Nash equilibrium or a super-strong Nash equilibrium does not always exist in general and it is NP-hard to check the existence of these equilibria. We consider an alternative notion of strong Nash equilibria and super-strong Nash equilibrium. Under such notions, we characterize the super-strong Nash equilibrium by Pareto-optimal strategy profiles. In the end, we show that it is NP-complete to find a manipulation that is strictly better for all members of the coalition. This result demonstrates a sharp contrast between weakly better-off outcomes and strictly better-off outcomes.
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