Don't know much about WW games, so you'll have to translate a bit. Was that last vampire a PC, so it was an immediate player-vs.-player setup?
It was Paris by Night. So if you want, an urban supernatural sandbox primarily populated with Vampires, with a whole set of factions, locales and particular NPCs to interact with (I had more than 120 fully fleshed out NPCs at one point, and more now with my NWoD reboot). The new PC basically shows up for the first time in the City, presents itself to the authorities of the region (like the Prince, Primogen etc for the Vampires) and then may decide to do whatever the hell he wants. Investigate the city's mysteries, blackmail other PCs or NPCs, carve a domain for himself within the city walls, and whatnot.
The last vampire in my example was a PC, yes. Assamites are assassins amongst vampires. "Antitribu" means that he was a rebel to his clan and a member of the Sabbat, the huge anti-Camarilla sect of the vampires in the WoD.
Otherwise how did the Patsis brothers fit into the rest of the group? (On what grounds were they working together or whatever?)
The rest of the players were Vampires. So right there, you had a Mage and a Bastet (werepanther, actually) who had no ties to their respective supernatural worlds, and were working against all odds together to hunt and kill Vampires as a character concept. So right off the bat I told them "you know what you guys are doing, right? These characters aren't going to survive this session." To which they laughed and said "Oh yes, sure". I mean, they didn't care. And at the time, I was very much in a "I'm completely neutral, and you guys can kill each others' characters so long as you're not asking me to take part in your fights" kind of frame of mind.
And so, the two of them killed all the other PCs of the table save one which basically killed them after a long chase scene in a very Matrix kind of way, decapitating one as he jumped off his bike and sword fighting on the busy highway the second one, finally getting him crushed by a passing truck, if I remember well, and finishing off with his blade.
But going back to your earlier post, in the pre-game discussions, do you ever see a character concept that's difficult or challenging? Did you have to exclude it, or did you find a way to include it?
No recent examples are really coming to mind, but talking in broad terms, if I identify a character idea I find out of place or wonder what to do with it, I'm going to say it point blank to the player. "What would you like me to do with this?" or "I don't quite see what we could do with that because of such and such. Or maybe if we tried that?" Usually doesn't take long to agree on basic character concepts that mesh well with each other.