Yes, games like that exist. There are many of them. You may wonder how such a game works, and how it differs from freeform improvisational collaborative storytelling.
To that end, I offer a modern, exemplary game of the type you are describing:
The game system that drives Hillfolk is called the DramaSystem, by noted designer Robin D. Laws, and it gives structure to a dramatic improvisation.
To see what such a game looks like in concrete, rather than abstract terms, check out The DramaSystem System Reference Document. The SRD is available under a couple of open licenses thanks to the success of the Kickstarter campaign. I have provided a link to the CC licensed version in the link.
The purpose of DramaSystem is to create experiences similar to good serial-drama TV, as seen in such shows as The Sopranos, The Wire, The Americans, or Sons of Anarchy. In order to accomplish this, the game defines characters less in terms of their skills and attributes and more in terms of their needs, wants, desires, and conflicts.
The game system is mostly based around the exchange of tokens based on whether a character was granted her request by another character or not. The system provides reasons to accede to requests when you might otherwise not, in order to gain the power to force requests of your own to be granted later on. In this way, the game mechanics prevent stagnation as everyone "digs in" on their positions.
Hillfolk is far from the only game in this space - but it is a complete, modern implementation of this kind of game that is freely available for your examination.