No, no. Ostracism was a free vote, conducted in the Assembly with a
quorum of voters present.
"Every year, the People's Assembly (ekklesia) was asked whether a
vote of ostracism should be held. If there was no clear majority, this
was the end of the matter. But if the people wanted to ostracize a
person, a day was set, typically two months later.
Every voter was given a potsherd (ostrakon) on which he wrote down
the name of a politician he believed to be potentially dangerous. (Or he
asked someone else to do the writing.) If a certain quorum was reached,
the politician who had received most votes was sent away from Athens.
The difference with an ordinary exile is that the man who was ostracized
remained a citizen, had to leave the city for a fixed period of ten
years, did not lose his possessions, and could be recalled - which
happened quite often."
http://www.livius.org/on-oz/ostracism/ostracism.html
Ed