Is there a closed-form expression for Shapley value of glove game? Suppose we have a coalition game with transferable utilities, with $m$ players having a right-handed glove and $n$ players having a left-handed glove. The value of a coalition is equal to the number of complete pairs of gloves in it. Then the Shapley value for a player with a right-handed glove is given by:
$\frac1{(m+n)!}\sum_{i=1}^{m+n}\sum_{j=0}^{\lfloor i/2 \rfloor-1}(m+n-i)!(i-1)!\binom{m-1}{j}\binom{n}{i-j-1}$
Is there a simple (or at least easy to evaluate) closed-form expression for this?
If not, perhaps an approximation for large m,n?
 A: I think you will find the article "The asymptotic shapley value for a simple market game" by Liggett, Lippman and Rumelt (2007) interesting and relevant for your question. 
They analyse a market game with $b$ buyers (in your case, this would be the $m$ players having a right-handed glove) who each seek to purchase 1 unit of an indivisible good form $s$ sellers (the $n$ players with a left-handed glove), each of whom has $k$ units to sell. I think that if $k=1$, this game coincides with the glove game. If $V(b,s)$ denotes the notes the Shapley value for a seller in this game, it turns out that $$V(b,s) = \frac{1}{b+s} \sum_{i=0}^{s+b-1} \sum_{2j > i} \frac{\binom{b}{j} \binom{s-1}{i-j}}{\binom{s+b-1}{i}} \quad . $$
Suppose $b, s \to \infty$, so that $b/ks = b/s \to \alpha$. If $\alpha = 1$, and we assume that $\frac{ks-b}{\sqrt{b+s}} \to u $, then $$V(b,s) \to \frac{k^{2}}{\sqrt{2 \pi}} \int_{0}^{\infty} \frac{x^{2}}{u^{2} + kx^{2}}e^{-x^{2}/2} dx \quad \text{if } u \geq 0. $$
There's a similar formula for the case in which $u \leq 0$. 
