More than one pirate faction to bid for, pirate factions that specializes in say, taking out enemy fleets(more like mercenaries), taking out enemy trade (standard piracy), and destroying enemy planets (warlord style pirates, expensive!). These three options would cost different amounts, and any player or team in the game could hire their own version of these pirates.
Say each team (even a team of 1) gets a pirate bid on a timer instead (instead of 1 overall timer, ever team has their own individual timer).
The team would then choose which pirate faction to bid for, the person on the team that bids the most wins control of the pirate faction that they were bidding on.
This allows for more pirates on the map, but the pirate fleets would be smaller and more like harassment forces instead of large fleets.
Depending upon the amount of money put up, the pirate fleet would be stronger or weaker. There could be a pirate fleet size cap, but this could be expanded with some research or capture of a pirate base.
In this way, people could still compete with each other, since pirate timers would expire all at the same time(and therefore pirate fleets would spawn at the same time, and everyone would be occupied with them, in theory). Instead of direct competition it would be more indirect competition. This would also be more balanced for multi player teams, since you would always be able to hire a mercenary pirate force, perhaps even to defend against another pirate force. The player who donates the most cash on the team would be the person that gives orders to these pirates.
Perhaps the capture of the local pirate base would give a team another pirate faction to hire a special "huge fleet of pirates" that they could hire at the most expensive price. In this way you could actually have people fighting over pirate bases and allow for another strategic objective a team can fight for on a map.
Perhaps this idea could bring about more pirates in large multiplayer matches and add to the feel of the game.