I used to be researching sleep, but now I'm going to switch focus to games and fun. First, I think it'll be useful to look at intrinsic vs. extrinsic rewards.
Intrinsic rewards are part of the game or experience. When you learn to ride a bike, all of a sudden you can ride a bike, and that's fun! Extrinsic rewards are outside of the experience. When you do well on a math test and get a good grade, that's an extrinsic reward; the test wasn't fun, but you get a prize afterward.
In short: intrinsic rewards work, extrinsic ones don't. Money (an extrinsic reward) sure doesn't. As usual, Eric Barker provides a nice summary. Recently, Dan Pink and others have expanded this "money doesn't motivate people" theory to "people want autonomy, mastery, and purpose."
In long: it's more nuanced than that; sometimes extrinsic rewards can be turned into intrinsic rewards by reframing the question: "will I?" instead of "I will." And it's possible that some extrinsic motivation actually works, especially if you use "integrated regulation of motivation", where you pick something that you don't enjoy, but you deeply value, and you approach it in your own way. Cal Newport has more. But this integrated-regulated-extrinsic motivation is unlikely to be found in games. Few people explicitly set out to master Halo because it complies with their core values.
So we're often left with games that are intrinsically fun, and "gamified things" that try to slap points and badges on top of otherwise not-fun things. Foursquare is a notable example of the latter, as is Epic Win.
A few game-related papers that point to the necessity of intrinsic motivation in games:
Bruckman (1999) offered the powerful and recurring "chocolate-covered broccoli" metaphor to represent games that combine boring drills with fun interludes.
Laschke and Hassenzahl (2011) express this succinctly in a workshop paper that contrasts patronage (going to a place over and over, becoming friends with the owner; intrinsically rewarding) with Foursquare mayorship (getting points when you "check in"). They conclude: "What is needed is a way to integrate single successes into a meaningful whole – a requirement, which is much better met by meaningful experiences than single rewards."
Habgood, in his doctoral thesis (2007) compared a version of a game where learning was integrated into gameplay with a version where it was external, concluding "... the integrated version is motivationally and educationally more effective than the extrinsic equivalent." (at least, that's what the abstract says. it's 250 pages long.)