This paper deals with a strategic issue in the stable marriage model with complete preference lists (i.e., a preference list of an agent is a permutation of all the members of the opposite sex).Given complete preference lists of all the men, a partial marriage, and complete preference lists of unmatched women, we consider the problem of finding preference lists of matched women such that the men-proposing Gale-Shapley algorithm applied to the lists produces a (perfect) marriage which is an extension of a given partial marriage. We propose a polynomial time algorithm for finding a desired set of preference lists, if these exist.We also deal with the case that complete preference lists of all the men and a partial marriage are given. In this case, we consider a problem of the existence of preference lists of all the women such that the men-proposing Gale-Shapley algorithm produces a marriage including a given partial marriage. We show NP-completeness of this problem.

SUMMARYThis paper deals with a strategic issue in the stable marriage model with complete preference lists (i.e., a preference list of an agent is a permutation of all the members of the opposite sex). Given complete preference lists of n men over n women, and a marriage µ, we consider the problem for finding preference lists of n women over n men such that the men-proposing deferred acceptance algorithm (Gale-Shapley algorithm) adopted to the lists produces µ. We show a simple necessary and sufficient condition for the existence of a set of preference lists of women over men. Our condition directly gives an O(n 2 ) time algorithm for finding a set of preference lists, if it exists.