The primogen council must determine how the city will be managed, particularly where hunting grounds are and granting turf and territory to clans. Granting turf is one way to manage large cities, because a prince and her sheriffs can't be everywhere at one, territory is granted to clans for feeding and haven purposes, and in return they keep their houses in order - watch for masquerade breaches, or threats like the sabbat or witch-hunters.
Checks and Balances
Thee primogen provide a check to the prince's power. The prince's behavior and decisions are monitored by the primogen, and their support of the prince's regime can be withdrawn if the prince steps over the line too many times.
Promote Clan Interests
The primogen can defend clan members from attack, or represent them in legal proceedings if they are accused of a crime. In addition to defending their own clans, they can also promote interests by suggesting territory rights, rights over businesses, and appointment to other key positions in the Camarilla.
The primogen counsel is not a close knit organization where everyone gets along and works together for the good of everyone. Most vampires are not that altruistic and counsel chambers can become a place of heated debate. Primogen may disagree with each other, or disagree with the Prince. When an issue comes up for a vote, the primogen may have already done some pre-emptives and called in boons (or offered them) to sway counsel members voting.
Most issues will go to a counsel vote, only the strongest and most dictatorial princes will go against the wishes of his counsel, and to do so is risky. The voting process and decision making can become extremely tangled, fraught with backbiting and favor trading.
Weapons of Choice
You May Not Sit
The counsel may decide to block a clan from taking its seat on the counsel, by simply stating that they do not recognize the authority of that person to speak for her clan. The Prince may even do the same.
Resources held by clans can be used to against others. The brujah may block access to certain institutions or feeding grounds that fall in his turf. On the flipside, primogen may point out shortcomings of other clans to manage their own resources, if masquerade breaches and other unsavory things are constantly occuring in the Brujah's territory, the Ventrue may suggest that someone else should take over, present it for a vote which could result in the loss and gain of resources.
This method is used to either force the counsel's attention on a topic, by not letting it go, or to prolong the discussion and thereby prevent a vote over a particularly tricky topic. Kindred only have the night hours to make a decisioin, and a clever primogen can be long winded enough to prevent the vote from occurring at all...perhaps so he or she can regroup or consult with her clan on the subject, or just to get things done before a decision is made. For instance, if the primogen are considering granting turf rights to an industrial district, a primogen may use filibustering to waylay the votes, so that she can gather her interests or present a better case for the acquisition of the territory.
That Old Gang-up of Mine
Primogen may coalesce together to thwart a movement by either the primogen or the prince. A proposition that goes to vote may be struck down with finality if a bloc on the primogen has decided (usually prior to the meeting) to block the proposition. This method may require some politicking on the primogen's part, to ensure the gang-up is successful.
A favored technique among toreador and ventrue, it involves getting harpy cooperation to watch a primogen closely, pointing out shortcomings that result in a loss of status. Primogen with low status will not have a strong influence over the council or even over their own clans.