Current New York law provides that if no one receives at least 40% of the vote for a New York city mayoral primary, a run-off primary is held. According to this story, the entrance of former Congressman Anthony Weiner into the Democratic race makes it somewhat likely that no one will get as much as 40% of the vote.
The run-off primary would be held on October 1, using the old-fashioned mechanical voting machines that have been outlawed for federal elections, if the legislature passes S.4088. The Senate passed it on May 6 but it has not yet had a hearing in the Assembly Elections Committee. Revival of the mechanical voting machines is opposed by Common Cause, the League of Women Voters, and other groups, according to this press release. However, one advantage of the lever voting machines is that they save time, because they avoid having to print hundreds of thousands of paper ballots.
Mayor Michael Bloomberg supports the lever machines; see this story. Anthony Weiner favors ranked-choice voting, at least for primaries. See his point #44, page 15, in his program.