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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
[1] Aumann R.J.: Values of markets with a continuum of traders. Econometrica 46, 611–646 (1975) · Zbl 0325.90082 · doi:10.2307/1913073
[2] Gul F.: Bargaining foundations of Shapley value. Econometrica 57, 81–95 (1991) · Zbl 0677.90011 · doi:10.2307/1912573
[3] Hart S., Mas-Colell A.: Bargaining and value. Econometrica 64, 357–380 (1996) · Zbl 0871.90118 · doi:10.2307/2171787
[4] Liggett T.M.: An invariance principle for conditioned sums of independent random variables. J Math Mech 18, 559–570 (1968) · Zbl 0181.20502
[5] Shapley L.S., Shubik M.: Pure competition, coalitional power, and fair division. Int Econ Rev 10, 337–362 (1969) · doi:10.2307/2525647
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