Sequential First-Price Auction with Randomly Arriving Buyers

Shulin Liu; Xiaohu Han

doi:10.21078/JSSI-2018-029-06

Artikel

Sequential First-Price Auction with Randomly Arriving Buyers

Shulin Liu und Xiaohu Han

Veröffentlicht/Copyright: 15. März 2018

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Informationen für Autor*innen

Aus der Zeitschrift Journal of Systems Science and Information Band 6 Heft 1

Abstract

In this paper we reanalyze Said’s (2011) work by retaining all his assumptions except that we use the first-price auction to sell differentiated goods to buyers in dynamic markets instead of the second-price auction. We conclude that except for the expression of the equilibrium bidding strategy, all the results for the first-price auction are exactly the same as the corresponding ones for the second-price auction established by Said (2011). This implies that the well-known “revenue equivalence theorem” holds true for Said’s (2011) dynamic model setting.

Keywords: sequential first-price auction; sequential second-price auction; dynamic market; symmetric Markov equilibrium; the difference equation

Supported by the National Natural Science Foundation of China (71171052)

Acknowledgements

The authors gratefully acknowledge the Editor and two anonymous referees for their insightful comments and helpful suggestions that led to a marked improvement of the article.

References

[1] Wolinsky A. Dynamic markets with competitive bidding. Review of Economic Studies, 1988, 55(55): 71–84.10.2307/2297530Suche in Google Scholar

[2] De Fraja G, Sákovics J. Walras retrouvé. Journal of Political Economy, 2004, 109(4): 842–863.10.1086/322087Suche in Google Scholar

[3] Satterthwaite M, Shneyerov A. Convergence to perfect competition of a dynamic matching and bargaining market with two-sided incomplete information and exogenous exit rate. Game and Economic Behavior, 2008, 63: 435–467.10.1016/j.geb.2008.04.014Suche in Google Scholar

[4] Said M. Sequential auctions with randomly arriving buyers. Games and Economic Behavior, 2011, 73(1): 236–243.10.1016/j.geb.2010.12.010Suche in Google Scholar

[5] Albano G L, Spagnolo G. Collusive drawbacks of sequential auctions. Journal of Public Procurement, 2011, 11: 139–170.Suche in Google Scholar

[6] McAfee R P, McMillan J. Auctions and bidding. Journal of Economic Literature, 1987, 25(2): 699–738.Suche in Google Scholar

Appendix

Recalling that δ = e^–ηΔ, p = λΔ, and q = ρΔ, we can find the limits of a, b, c, d, e, and f defined in Equation (6) and Equation (7) as Δ goes to zero, as shown below

limΔ→0a=limΔ→01−pqδ−(1−p)(1−q)δ(1−p)qδ=limΔ→01−λρΔ2e−ηΔ−(1−λΔ)(1−ρΔ)e−ηΔ(1−λΔ)ρΔe−ηΔ=limΔ→01−e−ηΔ(1−λΔ)ρΔe−ηΔ−limΔ→0λρΔ2e−ηΔ+e−ηΔ(−λΔ−ρΔ+λρΔ2)(1−λΔ)ρΔe−ηΔ=limΔ→0η(1−λΔ)ρe−ηΔ−limΔ→0λρΔ+(−λ−ρ+λρΔ)(1−λΔ)ρ=η+λ+ρρ.

limΔ→0b=limΔ→0p(1−q)(1−p)q=limΔ→0λΔ(1−ρΔ)(1−λΔ)ρΔ=limΔ→0λ(1−ρΔ)(1−λΔ)ρ=λρ.limΔ→0c=limΔ→0pδ(1−p)=limΔ→0λΔe−ηΔ(1−λΔ)=01=0.limΔ→0d=limΔ→0p(1−q)δ(1−p)q=limΔ→0λΔ(1−ρΔ)e−ηΔ(1−λΔ)ρΔ=limΔ→0λ(1−ρΔ)e−ηΔ(1−λΔ)ρ=λρ.limΔ→0e=limΔ→0p1−δp=limΔ→0λΔ1−e−ηΔλΔ=01=0.limΔ→0f=limΔ→0(1−p)δ1−δp=limΔ→0(1−λΔ)e−ηΔ1−e−ηΔλΔ=11=1.

Received: 2016-10-10

Accepted: 2017-2-7

Published Online: 2018-3-15

Published in Print: 2018-3-26

Sie haben derzeit keinen Zugang zu diesem Inhalt.

Artikel in diesem Heft

https://doi.org/10.21078/JSSI-2018-029-06

Schlagwörter für diesen Artikel

sequential first-price auction; sequential second-price auction; dynamic market; symmetric Markov equilibrium; the difference equation