Maximally representative allocations for guaranteed delivery advertising campaigns

Abstract

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.

Appendix

Proof of Theorem 1

The first order conditions, when $s_i^j >0$, come in the form

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

Thus,

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

(9)

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

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

(10)

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

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

(11)

Define

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

(12)

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

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

(13)

$$\begin{aligned} A^{j}&= \{i|y_i^j >0\}=\{i|p_i <p^{*j}\}. \end{aligned}$$

(14)

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

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

(15)