# Two Pricing Mechanisms in Sponsored Search Advertising

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review and Contribution of the Paper

## 3. Pricing Mechanisms

#### 3.1. Equilibrium in GSP and APR

**Definition 1**(Varian [7]) In a symmetric Nash equilibrium (SNE) bid prices satisfy

#### 3.2. Optimal Reserve Price in Sponsored Search Advertising

#### 3.2.1. GSP

**Lemma 1**In the lower bound of the locally envy-free equilibrium, the expected revenue rate to the search engine from the static generalized second-price auction (GSP) for sponsored search advertising is

**Theorem 1**If the value per click of each advertiser is a variable with IGFR, then (i) ${\Psi}_{A}(r)$ is a quasi-concave function; (ii) the optimal reserve price ${r}^{*}$ is given as the solution to the equation

#### 3.2.2. APR

**Lemma 2**In the locally envy-free equilibrium, the expected revenue rate to the search engine from APR for sponsored search advertising is

**Theorem 2**If each advertiser’s per-click value is a random variable with IGFR, then ${\Psi}_{P}(r)$ is quasi-concave and there exists a unique reserve price that maximizes the expected revenue rate. The optimal reserve price ${r}^{*}$ is given as the solution to the equation

## 4. Discussions

#### 4.1. Revenue Comparison

**Corollary 1**The revenue surplus of GSP over APR is given explicitly by

#### 4.2. Elevated Reserve Price

**Corollary 2**In APR mechanism selling multiple heterogeneous ad positions, the optimal reserve price should be set higher than the one in GSP.

**Example 1**There are $k=5$ sponsored links for sale. Bid arrival follows a Poisson distribution with mean $\lambda =20$. The per-click value of each advertiser is random and uniformly distributed within $[0,1]$. The CTRs are defined according to the empirical results in Brooks [2].

#### 4.3. Advertiser Population

**Corollary 3**In APR where the value distribution is IGFR, the optimal reserve price r is an increasing function of the arrival rate of advertisers λ.

## 5. Conclusion

## Appendix

**Proof of Lemma 1**Let ${R}_{A}(n,n\wedge k)$ be the expected average revenue rate from a GSP. We have

**Proof of Theorem 1**Taking the derivative of ${\Psi}_{A}(r)$ with respect to r leads to

**Proof of Lemma 2**Let n be the number of qualified bidders. When $n=1$, the search engine gains the expected revenue $E[{R}_{P}(1,1)]={R}_{P}(1,1)={\alpha}_{1}r$. For $1<n\le k$, the expected revenue is

**Proof of Theorem 2**Taking the derivative of ${\Psi}_{P1}(r)$ with respect to r, we obtain

**Proof of Corollary 1**Subtracting (16) from (14) gives

**Proof of Corollary 2**In light of Equation (1) the optimal reserve price satisfies $g({r}_{0}^{*})=1$. In Equation (17), $\theta (r)$ is clearly positive. Hence, $g({r}^{*})=1+\theta ({r}^{*})>1$, and we must have ${r}^{*}>{r}_{0}^{*}$ because of the assumption of IGFR.

**Proof of Corollary 3**We let $\theta (r,\lambda )$ replace $\theta (r)$ to underscore its dependence on λ. However, $g(r)$ is irrelevant to λ. The first-order condition (17) can be rearranged as $1-g(r(\lambda ))+\theta (r(\lambda ),\lambda )=0$. Differentiating with respect to λ in the both sides of the equation establishes

## References

- Google Inc. 2011 Annual Report on Form 10-K. Available online: http://www.sec.gov/Archives/edgar/data/1288776/000119312512025336/d260164d10k.htm (accessed on 1 April 2012).
- Brooks, N. The Atlas Rank Report: How Advertisement Engine Rank Impacts Traffic; Technical Report; Atlas Institute, University of Colorado: Boulder, CO, USA, July 2004. [Google Scholar]
- Yahoo! Search Marketing Blog. Reserve Prices, Minimum Bids No Longer Fixed at $.10 for Sponsored Search. Available online: http://www.ysmblog.com/blog/2008/02/26/minimum-bids/ (accessed on 1 February 2009).
- Google Adwords Help. What Ad Position Is the First Page Bid Related to? Available online: http://adwords.google.com/support/bin/answer.py?hl=en&answer=104241 (accessed on 12 February 2011).
- Google Adwords Help. How Much Do I Pay for a Click on My Ad? What If My Ad Is the Only One Showing? Available online: http://adwords.google.com/support/bin/answer.py?hl=en&answer=87411 (accessed on 12 February 2011).
- Edelman, B.; Ostrovsky, M.; Schwarz, M. Internet advertising and the generalized second price auction: Selling billions of dollars worth of keywords. Am. Econ. Rev.
**2007**, 97(1), 242–259. [Google Scholar] [CrossRef] - Varian, H.R. Position auctions. Int. J. Ind. Organ.
**2007**, 25(6), 1163–1178. [Google Scholar] [CrossRef] - Yang, W.; Feng, Y.; Xiao, B. Dynamic reserve price in sponsored search advertising. LIU Post, 2012; (working paper). [Google Scholar]
- Myerson, R.B. Optimal auction design. Math. Oper. Res.
**1981**, 6(1), 58–73. [Google Scholar] [CrossRef] - Riley, J.G.; Samuelson, W.F. Optimal auctions. Am. Econ. Rev.
**1981**, 71(3), 381–392. [Google Scholar] - Maskin, E.; Riley, J. Optimal Multi-Unit Auctions; Oxford University Press: Oxford, UK, 1989. [Google Scholar]
- Bulow, J.; Roberts, J. The simple economics of optimal auctions. J. Polit. Econ.
**1989**, 97(5), 1060–1090. [Google Scholar] [CrossRef] - Edelman, B.; Schwarz, M. Optimal auction design and equilibrium selection in sponsored search auctions. Am. Econ. Rev.
**2010**, 100(2), 597–602. [Google Scholar] [CrossRef] - Xiao, B.; Yang, W. A revenue management model for products with two capacity dimensions. Eur. J. Oper. Res.
**2010**, 205(2), 412–421. [Google Scholar] [CrossRef] - Feng, Y.; Xiao, B. A continuous-time yield management model with multiple prices and reversible price changes. Manage. Sci.
**2000**, 46(5), 644–657. [Google Scholar] [CrossRef] - Wang, R. Auctions versus posted-price selling. Am. Econ. Rev.
**1983**, 83(4), 838–851. [Google Scholar] - Yahoo! Search Marketing Help. Pricing and Minimum Bids. Available online: http://help.yahoo.com/l/us/yahoo/ysm/sps/faqs_all/faqs.html#pricing (accessed on 1 May 2009).
- Harris, M.; Raviv, A. A theory of monopoly pricing schemes with demand uncertainty. Am. Econ. Rev.
**1981**, 71(3), 347–365. [Google Scholar] - Wilson, R.B. Nonlinear Pricing; Oxford University Press: Oxford, UK, 1993. [Google Scholar]
- Roth, A.; Sotomayor, M. Two-Sided Matching; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- Lariviere, M.A.; Porteus, E.L. Selling to the newsvendor: An analysis of price-only contracts. M&SOM-Manuf. Serv. Op.
**2001**, 3(4), 293–305. [Google Scholar] - Ziya, S.; Ayhan, H.; Foley, R.D. Relationships among three assumptions in revenue management. Oper. Res.
**2004**, 52(5), 804–809. [Google Scholar] [CrossRef] - Lariviere, M.A. A note on probability distributions with increasing generalized failure rates. Oper. Res.
**2006**, 54(3), 602–604. [Google Scholar] [CrossRef]

© 2013 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Yang, W.; Feng, Y.; Xiao, B. Two Pricing Mechanisms in Sponsored Search Advertising. *Games* **2013**, *4*, 125-143.
https://doi.org/10.3390/g4010125

**AMA Style**

Yang W, Feng Y, Xiao B. Two Pricing Mechanisms in Sponsored Search Advertising. *Games*. 2013; 4(1):125-143.
https://doi.org/10.3390/g4010125

**Chicago/Turabian Style**

Yang, Wei, Youyi Feng, and Baichun Xiao. 2013. "Two Pricing Mechanisms in Sponsored Search Advertising" *Games* 4, no. 1: 125-143.
https://doi.org/10.3390/g4010125