Ok, so your players probably have hands, not paws, but I liked the alliteration.
The traditional roleplaying game, D&D, is very much structured in a specific way: the GM has a specific game/plot/monster that the players have to beat. In this way, D&D is structured much like a computer RPG, it just happens to be played with multiple players (and so can Baldur’s Gate 2, not to mention World of Warcraft).
There is value to this model; if you have one GM who is particularly good at plotting stories and taking care of all of the details of the world, and a lot of players without a knack for world-building, well, why not run a game like this? Labor gets divided up appropriately, and everybody gets to do what they’re good at.
Typically, however, everybody has something to offer to the story if given the chance, and 4 minds can probably come up with better ideas overall than just 1. This is why, typically (especially if I really trust the players), I prefer to run games that are much more player-driven. For me, good player-driven systems are those which have narrative elements built into them. The most obvious examples would be any game with a Fate point system. Such systems tend to give fate points to the player, and they can narrate something about the world that is unlikely but still possible by expending a Fate point. However, for me, the pinnacle of player-driven games is Apocalypse World. I understand that there are MORE player-driven games, but I find that the lack of a solid authority in completely player-driven games tends to leave most people feeling unsatisfied. Apocalypse World is the perfect balance: it lets the players help create the world and orient the plot, but doesn’t give them too much power to determine results.