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The asymptotic Shapley value for a simple market game. (English) Zbl 1173.91372
Authors’ abstract: 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 \([ks-b]/\sqrt{b+s}\) of the excess supply relative to the square root of the number of market participants.

91B26 Auctions, bargaining, bidding and selling, and other market models
91A12 Cooperative games
91A40 Other game-theoretic models
60F99 Limit theorems in probability theory
Full Text: DOI
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