The asymptotic shapley value for a simple market game

We consider the game in which b buyers each seek to purchase 1 unit of an indivisible good from s sellers, each of whom has k units to sell. The good is worth 0 to each seller and 1 to each buyer. Using the central limit theorem, and implicitly convergence to tied down Brownian motion, we find a closed form solution for the limiting Shapley value as s and b increase without bound. This asymptotic value depends upon the seller size k, the limiting ratio b/ks of buyers to items for sale, and the limiting ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${[ks-b]/\sqrt{b+s}}$$\end{document} of the excess supply relative to the square root of the number of market participants.