By this i take it that you mean election of the ephors and gerousia?
In answer to that, the apella made up of all spartan citizens over the age of thirty, would vote by acclamation (i.e. applause, shouting etc) and evidence for this comes from
Plutarch.
There were 5 ephors elected that represented 5 regions in Sparta. The Ephors could come from any social class (something that was condemed by Aristotle because poor men elected would be open to bribery) they held their position for one year and no one could be re-elected as an Ephor.
There were 28 members of the Gerousia (and 2 kings). To be a member you had to be over the age of 60, and anyone could be elected.
Plutarch states that "the best and deserving men past the age of 60 years" were chosen. This however, was only practised in theory, as those who were elected tended to come from the upper aristocratic classes. Once you were elected as a member of the Gerousia, you held that position for life. Both were elected via acclamation, and this represented a democratic process (as every spartiate could vote).
Plutarch describes the process of the Gerousia being elected:
"The assembly gathered and picked men were shut up in a nearby building,, where they could neither see or be seen, yet they could hear the shouts of those in the assembly. For In this instance, and many others, it was shouting that decided between competitors...whoever was met with the loudest shouting was the man elected".
I hope thats what you were after. If you were after who 'chose' the Spartan Kings, they werent actually chosen, they were determined via heredity.
