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I hope you guys find this an interesting question. Imagine you want to implement a system that rewards people for their contributions (regardless of quality or any external assessment). What is a good model, considering the following conditions:

  • It shouldn't reward people that contributes "excessively" only to gain whatever you're giving as reward. For example, if you're giving an award for 5 contributions, then you can't give that award again because of the sixth contribution.
  • It shouldn't be too complex to be computationally unfeasible (although, if you know of something complex but interesting, I'd like to know about it).

As a clarification, the situation is such that you want to give a single award (instead of a variety of them).

Maybe you know some established theory dealing with this kind of stuff. Let me know if that's the case.

By the way, I searched in Google Scholar but I didn't find anything relevant.

Thanks!

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I do not really think this can be considered a mathematical question until you're more specific about what you mean by "best." –  Qiaochu Yuan Jun 9 '12 at 18:35
    
@QiaochuYuan Don't be distracted by "best". After all, it's not an optimization problem. I just want to read good ideas about this. –  Robert Smith Jun 10 '12 at 3:44
    
Anyway, I changed 'best' to 'good' to avoid any confusion :-) –  Robert Smith Jun 10 '12 at 3:46
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1 Answer

Your reward function can follow the Erlang distribution for instance. If $c$ denotes the contribution, then the reward is $r(c)$ given as $$r(c) = r_{\max} \int_{x=0}^c \dfrac{\lambda^k x^{k-1} \exp(-\lambda x)}{(k-1)!} dx$$ where $r_{\max}$ is the maximum reward one can earn. This is the Erlang distribution. The parameters $\lambda$ and $k$ affect the rate of decay and the shape of the distribution respectively.

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Looks nice. Reading about it... :-) –  Robert Smith Jun 8 '12 at 4:02
    
Correct me if I'm wrong, but I think the idea is calculating the probability of having c contributions in a given period of time when I know the average number of contributions. Assuming this is the right interpretation, the benefit of using Erland distribution instead of an exponential distribution would be that using Erland I can vary the shape of my distribution to fit better my particular situation or is there other argument here?. Also, I think you were focusing on a situation in which there are several rewards, right?. So, the higher the probability, the greater the award. –  Robert Smith Jun 8 '12 at 4:58
    
Uhm, now I realize you meant that the time between contributions was Erland distributed instead of the number of contributions. Then the question to ask would be "Given a mean time between contributions, what is the probability of having k contributions in some time span?". Is that right?. –  Robert Smith Jun 8 '12 at 16:58
    
Why is everyone spelling A.K. Erlang's name with a d? –  Rahul Jun 9 '12 at 2:21
    
@RahulNarain I don't know. I think I read it here or somewhere else but I should have written Erlang. Seems like there has been some edition over here, though. –  Robert Smith Jun 10 '12 at 3:48
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