"My take on the ludology/narratology debate has always been that it's a clever and completely false dichotomy. If what you're into is talking about interactive entertainment, then it's endlessly fertile ground. If what you're into is making interactive entertainment, it's literarily meaningless."
-- Mark Barrett, game designer
So which is more important - game or story? Every instance of what is referred to in the quote as interactive entertainment has narrative elements - in some cases flapping like vestigial limbs while others have huge plot museums for you to wander through - all to provide an answer to why you are here.
The quote above raises an interesting point. Is plot essential to play? Does it matter if you're having fun grinding a level or three that you skip the exposition? Nobody stopped a game of Tetris because it broke their immersion – yet if you need that information to complete the game, you’d best have it available!
The priority that an audience places on mechanical interaction or completing challenges against that of story that helps you suspend your disbelief is more likely driven by a designer than by audience. Yet different players demand different things from games and a cohesive, entertaining story is often used as one of them.
This presents a paradox for tabletop roleplaying games. I've seen games fall flat when players reject the set up. If an event is unexpected and inconsistent, even if it follows the rule of cool, some players hit disbelief and utter “That’s just silly” or engage their right to choose and engage in mayhem upon the local village.
Equally, an alternate reality game stands or falls on exposition. If you lack a strong lead then something new and shiny will distract your audience promising hidden stories, special benefits or unique merch. While the interaction and challenges may be brilliant, without exposition you’re going nowhere fast.
Interactive entertainments in the vein of RPGs or ARGs require a greater investment of attention over time. Using a story linked to the game as a reward is common to video games, RPGs and ARGs. Some games implement this reward exclusively while others use it in tandem with formal play using rule mechanisms.
Ultimately it comes down to the design objectives of the game. If you’re looking at formalised play then story is a secondary concern. If you’re more interested in free-form play then story is suited as a framework and reward to participation. Determining what the players want then becomes important.