Appointment of the Mayor and Deputy Mayor is officially done via a vote but had traditionally been done through taking turns. When the council skipped Houlihan over for Mayor ( http://tinyurl.com/yej9l7k ) a few years ago Houlihan spent weeks worked up about the issue.
Recently, Barth was skipped over for Deputy Mayor and Houlihan happily accepted the post instead.
Earlier this month the council reviewed their policy on appointing the Mayor and Deputy Mayor. Here is the video of the discussion about the policy. You can't make this stuff up.