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Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $ [q:1,1,1,1..1,2,2,..2] $. I need help with calculating the Shapley value of a player with a weight of $2$ and a player with a weight of $1$ as a function of $a$,$b$ and $q$.

I know how to calculate the Shapely value in general, but I would like to get a simple closed form for the result. I tried to think about what happens if there are only players with weights of $1$ (simple scenario $ a=2k-1,b=0,q=k $ I think you will agree that $ \phi_{1}(v)=\frac{1}{a}=\frac{1}{2k-1} $. the simple calculation is $ \binom{2k-2}{k-1}(k-1)!(k-1)!=\frac{1}{2k-1}=\frac{1}{a} $ which is $ \frac{1}{2k-1} $. The calculation for a player of weight 1 (when there are 1s and 2s) would be a sum of choosing i players of weight 1 and $ \frac{q-i}{2} $ players of weight 2. similar calculation for a player of weight 2.

I tagged binomial coefficients because I thought they could be useful for counting the occurrences.

Thanks in advance, Mati

also posted here: https://math.stackexchange.com/posts/199080

Thanks in advance, Mati

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    $\begingroup$ "I would like to calculate the shapely value....." Then I think you should! $\endgroup$
    – Steven Landsburg
    Commented Sep 18, 2012 at 21:00
  • $\begingroup$ This site is for research. If there's no research angle to your question, it will fit better at math.stackexchange.com $\endgroup$
    – Gerry Myerson
    Commented Sep 18, 2012 at 22:46
  • $\begingroup$ There is a research angle, it's for my MSc thesis. but maybe I should post this question there too. thanks. $\endgroup$
    – Mati
    Commented Sep 19, 2012 at 11:29
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    $\begingroup$ If you do post it there, please put a link there to this question, and put a link here to that question. We don't want duplication of effort. $\endgroup$
    – Gerry Myerson
    Commented Sep 19, 2012 at 12:55
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    $\begingroup$ You are right, Gerry. I posted the link in the question. thanks again. $\endgroup$
    – Mati
    Commented Sep 19, 2012 at 13:18

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