Your question is difficult to answer without more information on what you are trying to sell. Most auction houses want a minimum estimated "value". If your piece is below say 100 USD, GBP or EUR, it is probably best to look at online auction houses.
If not, the variables are cost, clients and unsold items. If for instance, the mint is Lyon, a French auction house would have better clients, because chances are that they will concentrate on French mints. Otherwise, Greek and Roman coins are the dominating second choice (after the coins of their own country) for collectors anywhere, so there is little difference if the mint is e.g. Rome.
If your piece is an absolute topper, the auction house that is prepared to write up the coin (or even better, to put in on the front cover of the catalogue) is best. Otherwise, calculate the numbr of unsold classical coins in previous auctions.
Cost is a fixed item most of the time, but auction houses will negotiate over cost if they like the item and they think it will command a high price.
Peter