A rules light game such as poker or chess would give you a lot of leeway in designing a scoring system and implementing the social systems, but probably has an insufficiently complex game state to allow for a large team size while still minimizing redundancy. If you want to develop for large teams (which is almost required to create a difference between true democracy and a representative system), I would suggest a highly customizable, complex game such as Civilization 5, perhaps by allowing each player to control and receive data from an initial unit with socially selected ability to control cities and subsequently produced units within the team.