To me, sandra_nz’s idea is exactly the same as the OP’s idea.
How’s that? One teaches better spending habits, while one teaches eating habits which may or may not be better.
I don’t know if it’d teach them anything but it’d be interesting.
When I was a wee little shaver, Coke (as in Coca-Cola) was a rare treat, a few times a year. It came in bottles and we each got two pops. Since you never knew how many more months it’d be till you got a new one, it wasn’t consumed liberally.
I worked in a hotel and it was people not being able to control themselves in the moment was the idea behind the minibar
I don’t know if paying them is really necessary. I think the experience of still having ice cream left when others don’t should be lesson enough.
In order to answer this we really need your wife to post some of your other ideas here.
I voted ‘bad idea’ but only to be kind to you. Maybe this is brilliant compared to your other ideas.
That’s cold, man. 
Well, I did forget this: 
I can’t see how this experiment is going to teach them any significant life lesson. Delayed gratification isn’t a good all into itself, just like impulsiveness isn’t bad just because it is impulsive. I mean, I don’t overeat sweets, not because I’m trying to get a reward, but because I don’t want the negative consequences that follow from that behavior. You’re essentially trying to teach your kids the value of looking both ways before they cross the street by giving them money. But what will happen will you stop giving them money? Will they have come any closer to realizing that carelessly crossing the street could get you killed?
Have the kids shown a problem with practicing self-control and regulating how much they eat? An 11 and 9 year-old are old enough to know when they are being manipulated, and if you’ve already made a point of telling them that they lack willpower in the past, they’ll definitely know what’s going on. If they choose to gobble up all their ice cream sandwiches in one week, it probably has more to do with them not wanting to participate in this game than a lack of self-control. It wouldn’t take long for me to lose patience with someone who gave me something and then paid me not to use it.
I still have mine. And, no you can’t have any of it.
DECAYED gratification.
Sandra_nz has the right idea.
I disagree about the importance of teaching delayed gratification; it is important, otherwise every popup ad and credit card offer will cause trouble later in life.
If this is about financial habits, then use money. However, if this is about eating habits, then why use money?
This reminds me of the allowance agreement that John Rockefeller made with his son. It was like a contract and had stuff like, “The allowance will increase 10 cents per week, each week that you keep your money in order” and “Every dollar put into savings beyond 20% of the weekly allowance will be matched dollar-for-dollar.” It was basically a way to teach that the more money you have, the more money you can make.
The original idea is ok, but I think something that just used money would be a much better idea (i.e. Each week you don’t spend your allowance, you’ll get an extra $X.00 next week).