Maybe your community jar was too big and hard to fill? Maybe you just need a smaller jar that would pay off a little sooner. If they have to wait too long for the reward, sometimes it can feel unattainable, then what's the point? As far as getting them all to decide, they can just take turns choosing.
Giving them their own jar to fill, to me, is kind of a scary thing, as it would promote feelings and behaviors that they may not yet be old enough to handle (rivalry, jealousy, competition, the despair of being the loser, etc...) - feelings that may have long-lasting effects. In a larger classroom setting, maybe that's not such a big deal (they'd likely have friends to take their side), but in such a small group, I dunno... As their tutor for academics, are you sure that's a road you wanna travel? Just something to think about...
During the winter, when there's school and often unpredictable weather, we haven't been going on field trips. Last time I gave them a choice, they chose places to go, but the girl wanted to walk in a local park, and the boy said he didn't. He wanted to go to the ScienceCenter, and the girl didn't. We put the choices in a box and had the little one draw one out. Everyone agreed they'd go to whatever was chosen by chance. I chose the little boy, because no one could accuse him of cheating. He pulled out the girl's choice. Then the older boy said he wasn't going to the park. He absolutely refused! I lectured him about going back on his word. He had agreed to abide by whatever choice his brother drew. I thought this was very poor sportsmanship and said so. After lots of distressing discussion, with the older boy remaining adamant, I said I was giving up field trips and bought them gifts.
Then, shortly after that, for one of her homework assignments, I had the girl write a letter to me. It could be about anything. It was so she would practice writing a letter, addressing an envelope, etc. She did very well. But she asked me to reinstate field trips instead of gifts. I doubt they liked their gifts as much as a field trip, though I see they have used the sidewalk chalk several times in the driveway, and they used up the bubble liquid. Still, a trip is something to remember, and the gifts were just more toys.
So then the latest trouble over field trips. I felt bad for the girl missing out on fireworks, which she sets such great store by. I had her promise that, if I gave her a special trip, she'd do all her homework and work really hard for me, since this was extra, not earned by accumulating stones. It just seemed she always lost out with this stuff. And then she didn't do a sheet of homework! It's obvious she thought I'd still take her to the fireworks and ignore her lapse, and I decided if you say something is contingent on good work, and then you don't do it, the trip is canceled. Otherwise, she will never take me seriously.
So, with all this extra distress over the trips/gifts, I wish I had never started it. It's been no end of trouble and dissension, and the kids aren't working any harder than they used to, and the older boy doesn't even care whether he gets stones or not, though the younger boy is really into it, though whether he understands what the stones are for, I'm not sure.
I came up with the idea of separate smaller jars and separate trips when they are full, a fairer plan, I thought.
As or fostering competition, they are already very competitive, and I doubt I can change that. I can certainly forbid them gloating over who has the most stones, though maybe that would inspire them to earn more. And kids do compete. We have team sports in school and prizes for exceptional work, and not every kid gets to win. Life is competitive. Do we really want children to deny this reality?
The road I want to travel now is to give up the jars and stones completely. It has only caused more trouble than it's solved. If there's any trouble over the new plan, that'll be what I do next. I've had enough of the arguments and unhappiness over the trips.
