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Voting schemes are expressed as N person, single or multistage games in extensive form with ordinal preferences. Some of the voters can be indifferent to certain outcomes. The voters are not allowed to consult one another before arriving at the result. A majority rule with tie breaking is used. The voting schemes are labelled as binary or nonbinary depending on whether the number of alternatives in every ballot is two or more than two. We prove the principle of optimality for the voting problems and hence derive an algorithm for determining all the Nash strategies in which the voters restrict themselves to admissible strategies and the corresponding Nash values. In N stage binary games with strict preferences there is a unique Nash value which can be realized if a majority of the players vote according to the Nash strategy. Finally the Nash equilibrium strategies derived here have the important property that if the voters vote according to a Nash strategy, then they obtain an outcome which is often more favourable to the majority than an outcome obtained by other game strategies like the straightforward strategies. It is very rare that the Nash strategy will yield an outcome which is the least preferred outcome for any majority of voters.