Is your AEA Deputy the equivalent of an Equity Rep here or am I totally off on a tangent?
Dee, I don't know where "Here" is but to try to explain... An "Equity Rep" is usually the term for the AEA Business Representative, an employee of AEA (ie. gets paid for that job) manages contracts for the union, interprets rules for its members, etc.
The deputy is a volunteer member of a company, doesn't have the power to interpret the rules/contract but is charged with bringing those questions to the union. It assists in providing anonymity for the other members; an issue can be brought by a company member to the deputy, who then either assists in resolving the situation or more often, passes the issue up to stage management. It can often act as a filter for stage management. Usually it gets resolved at this level, but if not, the deputy can/should contact the Equity office supervising the contract, ie. NY, Chicago or LA (note: NOT company management, theatre mgmt, etc...) so that EQUITY can then contact management. A good deputy will help diffuse a situation early, often before it even needs to come to stage management; a good deputy will recognize the difference between a legitimate complaint ("The carpenter smells of tequila") vs. the not-union-issues complaint ("The carpenter smells of garlic").
However, ANY member is still free to contact their union should their be an issue but it is best to go first thru deputy, then stage management.
As far as the election, usually I try to make a little joke of it "and now our favorite time... DEPUTY ELECTION!" if I think it will go over well with the group. I ask for volunteers... If I don't get any, I remind them of their function, let them know that it is important to me that we are following the rules (so they know that I've got their back, they don't have an uphill battle, etc.) If still no takers then I invite nominations... I usually give a pass to someone if they recently deputied and had a bad experience or if they are brand new... (NB: usually, at this point someone goes ahead and nominates themselves...) However, if they are brand new and want to do it, I'll ask someone else to serve as deputy Deputy, basically someone to help the newbie (it is a great way for a new member to learn the rules and I like to encourage that, but it can be a bit trial by fire).
Also, I usually schedule the election just before the lunch break or end of day. It encourages people to move it along when they know that stalling is just cutting into their release. If they are delaying rehearsal, they tend to be more chatty, stalling, not as focused on the task, asking pointless questions, etc.