First, Second, Third, and All Pay Auctions- Strategies, and Revenues
Research Article
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First, Second, Third, and All Pay Auctions- Strategies, and Revenues

Newton Xu 1*
1 Uppingham School
*Corresponding author: newtonxu@icloud.com
Published on 16 September 2025
Journal Cover
AEMPS Vol.216
ISSN (Print): 2754-1177
ISSN (Online): 2754-1169
ISBN (Print): 978-1-80590-359-8
ISBN (Online): 978-1-80590-360-4
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Abstract

This study presents the optimum bidding strategy for first, second, third and all pay auctions in this article. Assuming that all bidders are rational, meaning that they would all want to make the most profit and behave the same, and there exists an equilibrium strategy that all bidders use, compute a mathematical model for different types of auctions. First, believe that there exists an equilibrium strategy that the strategy is linear, if not then we would try non-linear equilibrium strategies, it could be any polynomial or even exponential. Solving it, there will be a bid function where bidders get the maximum profit while the probability of winning is high. Next step, calculate the expected revenue for the auctioneer using our results from the equilibrium strategy. They are the same due to the revenue equivalence theorem. It is discovered that under certain conditions, different auction formats yield the same expected revenue for the seller.

Keywords:

First-Price, Second Price, Third price, All-Pay, Revenue Equivalence Theorem

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Xu,N. (2025). First, Second, Third, and All Pay Auctions- Strategies, and Revenues. Advances in Economics, Management and Political Sciences,216,29-38.

References

[1]. Bahr B, Fiveable. (2025) Symmetric games - (Game Theory) - Vocab, Definition, Explanations | Fiveable. Fiveable.me. Accessed August 7. https: //library.fiveable.me/key-terms/game-theory/symmetric-games

[2]. Jin X., Dan M., Zhang N., Yu W., Fu X., Das S.K. (2012) Game Theory for Infrastructure Security. Handbook on Securing Cyber-Physical Critical Infrastructure, pp.31-53.

[3]. Zhang H.  (2025) ECON20710 Lecture Auction as a Bayesian Game. Accessed August 11. https: //hanzhezhang.github.io/teaching/Chicago_ECON207/207sol_auction.pdf

[4]. EconPort. (2025) EconPort - Second-Price Sealed-Bid Auction. Econport.org. Accessed August 11. http: //www.econport.org/content/handbook/auctions/commntypes/secondpricesealed.html

[5]. Computational Thinking. (2025) Second Price Auction & Mechanism Design. YouTube. Published October 16. https: //www.youtube.com/watch?v=QJF5yR0ivuk

[6]. Professor Bergstrom. Notes on Expected Revenue from Auctions. https: //www.google.com/url?sa=t& source=web& rct=j& opi=89978449& url=https: //economia.uniroma2.it/master-science/economics/corso/asset/YTo0OntzOjI6ImlkIjtzOjM6IjQyMSI7czozOiJpZGEiO3M6NDoiMTExMiI7czoyOiJlbSI7TjtzOjE6ImMiO3M6NToiMjRhMDciO30%3D& ved=2ahUKEwjC4O2-nP2OAxUTha8BHSyIOU8QFnoECBYQAQ& usg=AOvVaw0glw6Pwe7E03Auz0jH5FPs

[7]. Dudnyk I.  ECON 302 -Microeconomic Theory II: Strategic Behavior IRYNA DUDNYK Lecture Notes. Expected Revenue in Auctions. https: //www.sfu.ca/~idudnyk/Auctions_Part2_Revenue_Equivalence.pdf

[8]. Pellizzoni R. (2008) Algorithmic Game Theory. Published March 28. http: //chekuri.cs.illinois.edu/teaching/spring2008/Lectures/scribed/Notes20.pdf

[9]. Levin J.  Notes on the Revenue Equivalence Theorem. https: //www.economics.utoronto.ca/damiano/ps426/RET-Levin-Notes.pdf

[10]. Munoz-Garcia F.  (2025) Revenue Equivalence Theorem. Accessed August 9. https: //felixmunozgarcia.com/wp-content/uploads/2017/08/revenue_equivalence_in_auction_formats.pdf

References

[1]. Bahr B, Fiveable. (2025) Symmetric games - (Game Theory) - Vocab, Definition, Explanations | Fiveable. Fiveable.me. Accessed August 7. https: //library.fiveable.me/key-terms/game-theory/symmetric-games

[2]. Jin X., Dan M., Zhang N., Yu W., Fu X., Das S.K. (2012) Game Theory for Infrastructure Security. Handbook on Securing Cyber-Physical Critical Infrastructure, pp.31-53.

[3]. Zhang H.  (2025) ECON20710 Lecture Auction as a Bayesian Game. Accessed August 11. https: //hanzhezhang.github.io/teaching/Chicago_ECON207/207sol_auction.pdf

[4]. EconPort. (2025) EconPort - Second-Price Sealed-Bid Auction. Econport.org. Accessed August 11. http: //www.econport.org/content/handbook/auctions/commntypes/secondpricesealed.html

[5]. Computational Thinking. (2025) Second Price Auction & Mechanism Design. YouTube. Published October 16. https: //www.youtube.com/watch?v=QJF5yR0ivuk

[6]. Professor Bergstrom. Notes on Expected Revenue from Auctions. https: //www.google.com/url?sa=t& source=web& rct=j& opi=89978449& url=https: //economia.uniroma2.it/master-science/economics/corso/asset/YTo0OntzOjI6ImlkIjtzOjM6IjQyMSI7czozOiJpZGEiO3M6NDoiMTExMiI7czoyOiJlbSI7TjtzOjE6ImMiO3M6NToiMjRhMDciO30%3D& ved=2ahUKEwjC4O2-nP2OAxUTha8BHSyIOU8QFnoECBYQAQ& usg=AOvVaw0glw6Pwe7E03Auz0jH5FPs

[7]. Dudnyk I.  ECON 302 -Microeconomic Theory II: Strategic Behavior IRYNA DUDNYK Lecture Notes. Expected Revenue in Auctions. https: //www.sfu.ca/~idudnyk/Auctions_Part2_Revenue_Equivalence.pdf

[8]. Pellizzoni R. (2008) Algorithmic Game Theory. Published March 28. http: //chekuri.cs.illinois.edu/teaching/spring2008/Lectures/scribed/Notes20.pdf

[9]. Levin J.  Notes on the Revenue Equivalence Theorem. https: //www.economics.utoronto.ca/damiano/ps426/RET-Levin-Notes.pdf

[10]. Munoz-Garcia F.  (2025) Revenue Equivalence Theorem. Accessed August 9. https: //felixmunozgarcia.com/wp-content/uploads/2017/08/revenue_equivalence_in_auction_formats.pdf

Cite this article

Xu,N. (2025). First, Second, Third, and All Pay Auctions- Strategies, and Revenues. Advances in Economics, Management and Political Sciences,216,29-38.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

About volume

Volume title: Proceedings of ICFTBA 2025 Symposium: Financial Framework's Role in Economics and Management of Human-Centered Development

ISBN: 978-1-80590-359-8(Print) / 978-1-80590-360-4(Online)
Editor: Lukáš Vartiak, Habil. Florian Marcel Nuţă
Conference website: https://2025.icftba.org/Galati.html
Conference date: 17 October 2025
Series: Advances in Economics, Management and Political Sciences
Volume number: Vol.216
ISSN: 2754-1169(Print) / 2754-1177(Online)

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