Today at GenCon I played a game called "King's Dilemna." King's Dilemna is a legacy board game where the players take the roles of a King's small council. Game mechanics wise you are presented with a scenario, given a choice, vote Yay/Nay/Pass, and reap the consequences and rewards of the choice of the council. Game play wise, you negotiate with your fellow players to achieve the objectives you want for your house, while pushing the kingdom into/away from chaos as fits your goals and whims at the moment.
In a lot of ways the game play is similar to games like Diplomacy. The point if managing your relationships with other players, figuring out what they want, helping or denying that as it fits your needs, and ultimately trying to achieve your own objectives. It is also something I've seen a lot of people try to put into their tabletop RPGs with varying degrees of success.
The fun thing is there is a key thing needed for social mechanics in games to work. That thing is conflict. Only, it is a special type of conflict. You don't just want violence - that just leads to fighting and PVP combat. No, if you want the social mechanics to come up you want conflict around a something specific and with specific requirements.
Big Enough To Matter. Small Enough to not be Final. Limited Availability
That line in bold is what you need to have a political game between your players. You also need it for a political game between players and NPCs but in that all the players are on the same side. The key thing here is you really need all three.
The reward has to be big enough that it is enticing to all, or at least most, parties involved. This can be the PCs, it can be PCs and NPCs, but you need enough that groups or individuals want the reward.
At the same time, the reward needs to be small enough that people aren't just going to murder the competition. If you put a kingdom's crown up for grabs, that is a big enough prize that a lot of people will just murder the competition for it. If you want that it is fine, but unless you're ok with a player sitting out because they're dead it is not something to do lightly. So it needs to be important, but not so important the value in having it trumps the consequences for more absolute solutions. You want people wheeling and dealing, not slitting and slicing.
Finally, the reward needs to be limited in availability. If you have 4 factions, and enough reward for all four there is no conflict. If you have 4 factions and enough reward for 3, then you will quickly end up with a 3 man alliance and someone on the outside. But if you have 4 factions and enough for 2 or only 1? Now you have a game.
Next to these three things, the next part for a longer term campaign is to have another thing that follows up after or multiple rewards at the same time. Not all rewards have to be equally enticing to all parties, but they should have some. Let people build relationships. Let them make deals. Let them break those deals. Let them make enemies, allies, etc. Let them play the game.
But one last thing: you can't force people to do it. Present the situation, present the reward, but if the players decide to work as a group or try to make it work with all the NPCs don't force them out of it. Actions and consequences are fine, but forcing a political game can be bad for the game.
Present, show, enjoy....don't force. And have fun with it!