Contributions by Maskin (1977) and Williams (1984b) have established that any social choice correspondence satisfying monotonicity, no veto power, and having at least three participants is Nash implementable, under the proviso that the number of social alternatives is finite and that the social choice correspondence satisfies citizen sovereignty. For an alternative set of arbitrary size, however, Williams' proof requires an additional restriction on the nature of the alternative set. In this paper, I make improvements in two important aspects of their Nash implementation theorems: a significant reduction in the strategy space and a proof for an arbitrary alternative set. In Maskin's and Williams' game forms, each participant announces a preference profile of all participants and also a socially optimal alternative with respect to the announced preference profile. I prove Maskin's theorem using a much smaller strategy space with respect to the preference announcements. Namely, each participant announces only two participants' preferences (i.e., his own preferences and his successor's preferences), an alternative that is not necessarily optimal, and a positive integer not exceeding the number of participants. With this specification of the strategy space, I confirm the theorem for an arbitrary size of the alternative set.