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Maximally representative allocations for guaranteed delivery advertising campaigns

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There are around 400 advertising networks that match opportunities for “display” advertising, which include banner ads, video ads and indeed all ads other than text-based ads, on web pages and candidate advertisements. This is about a \({\$}25\) billion business annually. The present study derives a method of pricing such advertisements based on their relative scarcity while ensuring that all campaigns obtain a reasonably representative sample of the relevant opportunities. The mechanism is well-behaved under supply uncertainty. A method based on the mechanism described in this paper was implemented by Yahoo! Inc.

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  1. The cleanest treatment of the inverse elasticity rule is Varian (1985).

  2. The problem of display advertising is discussed in Babaioff et al. (2008), Boutilier et al. (2008), Contantin et al. (2009) and Ghosh et al. (2009).

  3. The problem of orphan categories arises in the literature on expressive bidding, exemplified by Boutilier et al. (2008) and Agarwal et al. (2007).

  4. The role of \(r_{i}\) is as a floor or reserve price. It is the minimum that should ever be accepted, and can be thought of as the price that obtains by auctioning the impression in an exchange or the value of running a “house ad” (an ad for the website’s own content) or a public service advertisement.

  5. Otherwise merging two similar supply pools changes the objective function. Consider merging two supply pools, with the same relative shares. The change in the objective function, using the proportional weights, is \(\frac{1}{x_1 }\left( {\frac{x_1 }{X}-\frac{y_1}{Y}} \right) ^{2}+\frac{1}{x_2 }\left( {\frac{x_2 }{X}-\frac{y_2}{Y}} \right) ^{2}-\frac{1}{x_1 +x_2 }\left( {\frac{x_1 +x_2}{X}-\frac{y_1+y_2}{Y}} \right) ^{2}=\frac{x_1 x_2}{Y^{2}(x_1 +x_2)}\left( {\frac{y_2}{x_2}-\frac{y_1}{x_1}} \right) ^{2}=0\).

  6. Beck and Milgrom (2012) also use an alternative randomized allocation, based on the gap between first and second prices, to improve efficiency of the allocation.

  7. The contribution of Ghosh et al. (2009) is to show how to rationalize a set of such campaigns that might conflict with each other; in particular it could be necessary to divert inventory that one campaign loses to another campaign. This in particular arises whenever \(1>\sum _j {\frac{1}{V^{j}}\left( {\frac{p^{*j}Y^{j}}{\sum _k {s_k^j x_k } }} \right) \left( {1-{p_k }\left. /\right. {s_k^j p^{*j}}} \right) } \).


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Correspondence to R. Preston McAfee.

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The authors are grateful to Yahoo! Inc, where this work was performed.



Proof of Theorem 1

The first order conditions, when \(s_i^j >0\), come in the form

$$\begin{aligned} 0=-s_i^j V^{j}\left( {\frac{x_i }{\sum \limits _k {s_k^j x_k } }-\frac{y_i^j }{Y^{j}}} \right) \frac{\sum \limits _k {s_k^j x_k } }{x_i }+p_i -s_i^j \alpha ^{j}, \,\text{ or }\, y_i^j =0\quad \text{ and }\, \frac{\partial L}{\partial y_i^j}\ge 0. \end{aligned}$$


$$\begin{aligned}&\frac{s_i^j y_i^j }{Y^{j}}=\left( {\frac{s_i^j x_i }{\sum \limits _k {s_k^j x_k } }} \right) \left( {1-\frac{p_i/s_i^j -\alpha ^{j}}{V^{j}}} \right) \, \text{ or } \quad y_i^j =0\nonumber \\&\quad \text{ and }\nonumber \\&0\ge \left( {\frac{s_i^j x_i }{\sum \limits _k {s_k^j x_k } }} \right) \left( {1-\frac{p_i /s_i^j -\alpha ^{j}}{V^{j}}} \right) . \end{aligned}$$

Let \(A^{j}=\{i|y_i^j >0\}.\) Summing (9) over \(i\in A^{j},\) and solving for \(\alpha ^{j}\) we obtain

$$\begin{aligned} \alpha ^{j}=\frac{V^{j}\sum \limits _k {s_k^j\,x_k} +\sum _{i\in A^{j}} {x_i \, p_i} }{\sum \limits _{i\in A^{j}} {s_i^j\,x_i}}-V^{j}. \end{aligned}$$

Substituting the value of \(\alpha ^{j}\) into (9), we obtain a solution for \(y_i^j \):

$$\begin{aligned} \frac{s_i^j y_i^j }{Y^{j}}&= \left( {\frac{s_i^j x_i }{\sum \limits _k {s_k^j x_k}}} \right) \left( {1-\frac{p_i /s_i^j -\alpha ^{j}}{V^{j}}} \right) \nonumber \\&= \frac{1}{V^{j}}\left( {\frac{s_i^j x_i }{\sum \limits _k {s_k^j x_k } }} \right) \left( {\frac{V^{j}\sum \limits _k {s_k^j x_k } +\sum \limits _{i\in A^{j}} {x_i p_i } }{\sum \limits _{i\in A^{j}} {s_i^j x_i } }-p_i /s_i^j } \right) \end{aligned}$$


$$\begin{aligned} p^{*j}=\frac{V^{j}\sum \limits _k {s_k^j x_k } +\sum \limits _{i\in A^{j}} {x_i p_i } }{\sum \limits _{i\in A^{j}} {s_i^j x_i}} \end{aligned}$$

Now, note that \(y_i^j =0\) if and only if \(p_i \le s_i^j p^{*j}\). Thus we have:

$$\begin{aligned} \frac{s_i^j y_i^j }{Y^{j}}&= \left( {\frac{s_i^j x_i }{\sum \limits _k {s_k^j x_k } }} \right) \frac{Max[0,\quad p^{*j}-p_i /s_i^j ]}{V^{j}}, \text{ and }\end{aligned}$$
$$\begin{aligned} A^{j}&= \{i|y_i^j >0\}=\{i|p_i <p^{*j}\}. \end{aligned}$$

This solution delivers quantities as a function of prices. The prices are endogenized as follows. Recall that there is an exogenous price \(r_{i}\) at which the guaranteed inventory can be sold to the outside. Then the prices are given by complementary slackness—either

$$\begin{aligned} p_{i}=r_{i}\quad \text{ and } \quad x_i \ge \sum _j {y_i^j} \quad \text{ or } \quad p_{i} > r_{i} \quad \text{ and } \quad x_i =\sum _j {y_i^j}. \end{aligned}$$

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McAfee, R.P., Papineni, K. & Vassilvitskii, S. Maximally representative allocations for guaranteed delivery advertising campaigns. Rev Econ Design 17, 83–94 (2013).

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