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2021 | OriginalPaper | Chapter

2. First-Price Auctions

Authors : Pak-Sing Choi, Felix Munoz-Garcia

Published in: Auction Theory

Publisher: Springer International Publishing

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Abstract

In this chapter, we analyze first-price auctions, where every bidder privately observes his valuation for the object and submits his bid to the seller (e.g., in a sealed envelope). The seller ranks all bids, selects as a winner the bidder who submitted the highest bid to pay the bid he submitted. All other bidders pay zero and not receive the object.

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previous chapter Second-Price Auctions

next chapter First-Price Auctions: Extensions

For a more detailed historical account of Dutch auctions, among other auction formats, see Lucking-Reiley (2000) and Karpowicz (2010).

Recall that \(b_{i}\left ( b_{i}^{-1}\left ( x_{i}\right ) \right ) =x_{i}\). Differentiating both sides with respect to x _i, and applying the Chain rule, we obtain \(\frac {\partial b_{i}\left ( b_{i}^{-1}\left ( x_{i}\right ) \right ) }{\partial b_{i}^{-1}\left ( x_{i}\right ) }\frac {\partial b_{i}^{-1}\left ( x_{i}\right ) }{\partial x_{i}}=1\), which we can rearrange to yield \(\frac {\partial b_{i}^{-1}\left ( x_{i}\right ) }{\partial x_{i}}= \frac {1}{b_{i}^{\prime }\left ( b_{i}^{-1}\left ( x_{i}\right ) \right ) }\).

Recall that, when integrating by parts, we consider two functions g(x) and h(x), such that (gh)^′ = g ^′ h + gh ^′. Integrating both sides yields \(g(x)h(x)=\int g^{\prime }(x)h(x)dx+\int g(x)h^{\prime }(x)dx\). Reordering this expression, we find \(\int g^{\prime }(x)h(x)dx=g(x)h(x)-\int g(x)h^{\prime }(x)dx\). At this point, we can apply integration by parts in our auction setting by defining g ^′(x) ≡ (N − 1)F(x)^N−2 f(x) and h(x) ≡ x, so we obtain the given results.

Karpowicz, M. (2010). Designing auctions: A historical perspective. Journal of Telecommunications and Information Technology, 3, 114–122.

Lucking-Reiley, D. (2000). Vickrey auctions in practice: From nineteenth-century Philately to twenty-first-century e-commerce. Journal of Economic Perspectives, 14(3), 183–192.CrossRef

Title: First-Price Auctions
Authors: Pak-Sing Choi
Felix Munoz-Garcia
Publisher: Springer International Publishing
Book: Auction Theory
Print ISBN: 978-3-030-69574-3

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

Copyright Year: 2021
DOI: https://doi.org/10.1007/978-3-030-69575-0_2