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Erschienen in:

2021 | OriginalPaper | Buchkapitel

1. Second-Price Auctions

verfasst von : Pak-Sing Choi, Felix Munoz-Garcia

Erschienen in: Auction Theory

Verlag: Springer International Publishing

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Abstract

We start our analysis of auction formats with the second-price auction, where the winning bidder does not pay the bid he submitted but, instead, the second-highest bid. This auction format is relatively easy to analyze, requiring limited mathematical steps, and thus we believe that it can serve as a first approximation to bidding behavior for non-technical readers. Examples of this auction format include eBay sales, auctioning radio spectrums, and the pricing that search engines, such as Google, Bing, or Yahoo, use to sell keyword-based advertising.

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Nächstes Kapitel First-Price Auctions

Recall that strategy s _i weakly dominates another strategy \( s_{i}^{\prime }\) if player i’s utility satisfies \(u_{i}(s_{i},s_{-i})\geq u_{i}(s_{i}^{\prime },s_{-i})\) for every strategy profile s _−i that his rivals choose and \(u_{i}(s_{i},s_{-i})>u_{i}(s_{i}^{\prime },s_{-i})\) for at least one strategy profile s _−i. For more details, see the Game Theory Appendix at the end of the book, which provides a list of game theoretic tools used throughout the book.

Intuitively, expression

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finds the highest bid among all bidders different from bidder i, j ≠ i . Alternatively, h _i can be written more explicitly as \(h_{i}=\max \{b_{1},b_{2},\ldots ,b_{i-1}, b_{i+1},\ldots ,b_{N}\}\), where we find the highest bid among all N bidders except for bidder i (note that we wrote everyone’s bid but i’s, b _i).

More generally, if K ≥ 2 bidders are tied submitting the highest bid, and the auctioneer randomly assigns the object to any of them, each of these bidders earns an expected payoff of \(\frac {1}{K}(v_{i}-h_{i})\).

Recall that, in the exponential distribution, λ represents the rate parameter, describing how quickly the decay of the function is. In addition, the expected value of the exponential distribution, \(E[x]=\frac {1}{\lambda }\) , and its variance, \(Var[x]=\frac {1}{\lambda ^{2}}\), are both decreasing in λ, implying that bidders’ expected valuation is more concentrated around the lower bound.

Recall that a strategy profile is a Bayesian Nash Equilibrium if every player chooses a best response (in expectation) given his rivals’ strategies, s _−i, and given his type. For a more formal definition, see the Game Theory Appendix at the end of this book. For a more detailed presentation, see Tadelis (2013, Chapter 12), and for examples of how to apply this game-theory tool to several games, see Munoz-Garcia and Toro-Gonzalez (2020, Chapter 7).

Augenblick, N. (2016). The sunk-cost fallacy in penny auctions. The Review of Economic Studies, 83(1), 58–86.CrossRef

Blume, A., & Heidhues, P. (2004). All equilibria of the Vickrey auction. Journal of Economic Theory, 114(1), 170–177.CrossRef

Che, Y. K., & Gale, I. (1998). Standard auctions with financially constrained bidders. The Review of Economic Studies, 65(1), 1–21.CrossRef

Graham, D. A., & Marshall, R. C. (1987). Collusive bidder behavior at single-object second-price and English auctions. Journal of Political Economy, 95(6), 1217–1239.CrossRef

Hendricks, K., & Porter, R. H. (1989). Collusion in auctions. Annales d’Êonomie et de Statistique, 15/16, 217–230.

Munoz-Garcia, F., & Toro-Gonzalez, D. (2020). Strategy and game theory, practice exercises with answer keys (2nd ed.). Springer-Nature.

Sayman, S., & Akçay, Y. (2020). A transaction utility approach for bidding in second-price auctions. Journal of Interactive Marketing, 49, 86–93.CrossRef

Tadelis, S. (2013). Game theory, an introduction. Princeton, NJ: Princeton University Press.

Titel: Second-Price Auctions
verfasst von: Pak-Sing Choi
Felix Munoz-Garcia
Verlag: Springer International Publishing
Buch: Auction Theory
Print ISBN: 978-3-030-69574-3

Electronic ISBN: 978-3-030-69575-0

Copyright-Jahr: 2021
DOI: https://doi.org/10.1007/978-3-030-69575-0_1

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