At my new job, on Fridays, the boss of the tutoring center gives the students a break from writing by letting them watch a movie. The boss couldn't bring a movie because he had to go to a meeting. So, I brought a TV series. I brought a TV series called Gargoyles. It's a classic cartoon that I used to watch back when I was little. I loved the story when I was in elementary school and I still love it now.
The students took a vote between Real Steel, Avatar: The Last Airbender, and Gargoyles. They chose Gargoyles, which is partly due to my impassioned description of the story. Ahead of time, I told the students that if they hadn't finished writing their paragraph, they could not watch the TV series with the rest of the class.
So, its time to watch the TV series. We moved the tables, I set up the projector and the DVD, and the student who had not finished his paragraph wasn't budging. So, I decided that I wouldn't play the DVD. That's when the whole class tells him to get out and finish his paragraph. So, he leaves the classroom and finishes it.
I was proud of this event for two reasons: #1 I got the students hooked on a TV series that I used to watch when I was close to their age. #2 I got a student who is normally pretty damn stubborn to comply. There's a lesson to be learned here, but it seems much easier to express than it is to implement. You find out what the class wants, only give it to those students who have completed the necessary academic requirements (which you must have decided in advance), and pause the distribution of the reward for the entire class if a student refuses to cooperate. Why pause it for the whole class? Well, because they'll pressure a maverick student into complying with the terms of the reward.
In a regular classroom of 20ish students, is it feasible to give a reward at the end of each week for completing certain academic requirements? Possibly, but at this point, I'm not sure what possible rewards their are and how exactly the conditions should be set. Depends on the situation at hand I guess. Do you have the time for it? Is it a kind of reward that the class would look forward to each week? Can you only attach it to a single assignment or could you attach it to multiple assignments? Which assignments? Those are some questions that would help you figure out whether such a reward system is feasible.