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01-09-2012

On the optimal number of advertising slots in a generalized second-price auction

Authors: Alex Kim, Subramanian Balachander, Karthik Kannan

Published in: Marketing Letters | Issue 3/2012

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Abstract

In search advertising, a search engine uses a generalized second-price auction to sell advertising slots adjacent to search results on its webpage. In this paper, we study an interesting question related to the design of the generalized second-price auction: how should a search engine strategically decide on the number of advertising slots? To answer this question, we analyze the implication of varying the number of slots in a base model in which the click-through rates are assumed to be independent of the number of slots. When deciding the number of slots, we find that a search engine’s profit is based on two counteracting factors: the incremental clicks from an extra slot and the influence of the extra slot on advertisers’ payments per click. Our analysis characterizes the conditions for optimality of the number of slots and the implications of different distributions for advertiser valuations. We also extend the base model to allow for attraction and cannibalization of clicks from existing slots by new ad slots and show how such effects affect the optimal number of slots. Our overall results show that search engines need to optimize the number of ad slots offered for auction in order to maximize profit.

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An alternative research stream develops empirical models of advertiser, consumer, and search engine behavior using consumer click data. Rutz and Bucklin (2007) and Ghose and Yang (2009) present qualitative response models of the relationship between various search ad metrics such as click-through rates, conversion rates, ad ranking, or position and cost per click. Yao and Mela (2011) develop a dynamic structural model of advertisers’ bidding behavior and consumers’ clicking behavior to empirically evaluate the impact of search results page design, auction pricing policy and advertiser bidding process on search engine revenues.

The maximum number of ad slots varies across search engines. For example, Bing has a maximum of nine ad slots, whereas Google has 11 slots.

Further, if advertisers expect that the publisher would use their bids to decide on the number of slots, they may bid strategically, thus making implementation difficult.

Our results are also applicable to web publishers that import and display the ads from search advertising service providers, e.g., NYTimes.com imports ads from Google.

Advertisers may participate as long as they see an opportunity to make a profit which will typically mean at least K + 1 advertisers would participate in the auction when K < N.

The superscript again accounts for the total number of ad slots available.

Note that our results do not require all N potential advertisers participate in the auction. It is sufficient if the top K + 1 advertisers in terms of v _i participate in an auction with K slots. Thus, our analysis allows for the number of bidders to depend on K.

The function f(x) is said to be log-concave if ln(f(x)) is concave. If f(x) is log-concave, so is 1 − F(x), hence the hazard rate is increasing in x, meaning IHR.

When the shape parameter is less than 1, the appendix shows that J(v) is always negative.

From Proposition 3 in Balachander et al. (2009), \( p_k^K = \frac{1}{{\alpha_k^K{\chi_{{g(k)}}}}}\left\{ {\sum\nolimits_{{j = k}}^K {{v_{{j + 1}}}\left( {\alpha_j^K - \alpha_{{j + 1}}^K} \right)} } \right\} \) and \( p_k^{{K + 1}} = \frac{1}{{\alpha_k^{{K + 1}}{\chi_{{g(k)}}}}}\left\{ {\sum\nolimits_{{j = k}}^{{K + 1}} {{v_{{j + 1}}}\left( {\alpha_j^{{K + 1}} - \alpha_{{j + 1}}^{{K + 1}}} \right)} } \right\} \). Note that \( \alpha_j^K = \alpha_j^{{K + 1}} \), for 1 ≤ j ≤ K, \( \alpha_{{K + 1}}^K \) = 0, \( \alpha_{{K + 1}}^{{K + 1}} \) > 0, and \( \alpha_{{K + 2}}^{{K + 1}} \) = 0. Then, we have \( p_k^K - p_k^{{K + 1}} = \frac{{\alpha_{{K + 1}}^{{K + 1}}}}{{\alpha_k^K{\chi_{{g(k)}}}}}\left\{ {{v_{{K + 1}}} - {v_{{K + 2}}}} \right\} > 0 \).

For example, with a uniform distribution on valuations over [0,1], Edelman and Schwarz (2006) would suggest a minimum bid of \( {b_{{\min }}} = \frac{1}{2} \), which results from solving \( v = \frac{{1 - F(v)}}{{f(v)}} \). Note that b _min depends on the range of valuations.

We skip the details of equilibrium analysis for equilibrium bids and profits in the second stage. See Balachander et al. (2009) for the details of equilibrium analysis

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Title: On the optimal number of advertising slots in a generalized second-price auction
Authors: Alex Kim
Subramanian Balachander
Karthik Kannan
Publication date: 01-09-2012
Publisher: Springer US
Published in: Marketing Letters / Issue 3/2012
Print ISSN: 0923-0645
Electronic ISSN: 1573-059X
DOI: https://doi.org/10.1007/s11002-012-9193-2

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