×

zbMATH — the first resource for mathematics

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.

MSC:
91B26 Auctions, bargaining, bidding and selling, and other market models
91A12 Cooperative games
91A40 Other game-theoretic models
60F99 Limit theorems in probability theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aumann R.J.: Values of markets with a continuum of traders. Econometrica 46, 611–646 (1975) · Zbl 0325.90082
[2] Gul F.: Bargaining foundations of Shapley value. Econometrica 57, 81–95 (1991) · Zbl 0677.90011
[3] Hart S., Mas-Colell A.: Bargaining and value. Econometrica 64, 357–380 (1996) · Zbl 0871.90118
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.