Mathematics and advertising

Category: Business, Economics, and Marketing.

Fields of Study: Fields of Study: Algebra; Data Analysis and Probability; Number and Operations.

Summary: Mathematics is used to weigh the costs and gains of advertising and to profile and target consumers.

Advertising delivers product information from suppliers to consumers—suppliers may be manufacturers, hospitals, software developers, educators—and is critical to the success of a business in marketing development. Advertising media may be traditional (such as television, newspapers, and posters) or technological (via Internet and e-mail), as well as commercial (to sell products for profit) or noncommercial (in political campaigns or for religious purposes). The annual advertising cost in the United States amounts to more than $100 billion.

Advertising includes two stages: the planning stage for marketing strategies, whose goal is business development, and the analysis stage of cost analysis involved with the forms and the contents of communication between suppliers and potential customers. Mathematics and statistics play critical roles in both stages of advertising.

Market Shares

In the planning stage, the analysis of market shares for advertising necessitates matrix operations and multivariate probability inequalities to portray the dynamics of market shares over time. The following is an example of matrix operations, which bridge advertising with market shares. Consider the market shares of General Motors (GM) and Ford in the U.S. automobile industry. Assume that the current market shares distribute as follows:

General Motors: 21%

Ford: 17%

Other Manufacturers: 62%

If GM starts an advertising campaign with the goal of increasing the market share to 29% in three years, GM may count on customers to switch from Ford or other manufacturers to GM. However, in reality, some of the GM customers may switch to Ford or to other manufacturers.

Let a₁, a₂, a₃ be the percentages of original GM users who, at the end of the advertising campaign, remain with GM, who switch to Ford, and who switch to other manufacturers, respectively. Let b₁, b₂, b₃ be the percentages of original Ford users who switch to GM, who remain with Ford, and who switch to other manufacturers, respectively. Let c₁, c₂, c₃ be the percentages of the other customers who switch to GM, who switch to Ford, and who remain with their manufacturers, respectively. Then, the market shares x_GM, x_Ford, and x_Others at the end of the three years are determined by the following simple matrix equation:

If GM intends to increase x_GM to 29%, GM should advertise specifically to different groups of customers. This is mathematically equivalent to manipulating the elements in the 3×3 matrix above within plausible ranges of the elements.

The foregoing scenario is a simplified example to illustrate the role of matrix operations in advertising. In reality, the story is more complex. For example, the 3×3 matrix above will become an n × n matrix, where n is the number of competing suppliers in the market. Also, the stochastic feature of the supply-demand market, the market shares, and the corresponding elements for the n × n matrix change constantly under the influence of the advertising campaign.

Thus, it is more appropriate to treat the market shares as a vector consisting of random variables. In this case, one of the convenient approaches to evaluating the market shares is the method of multivariate probability inequalities in conjunction with the construction of Hamilton-type circuits.

Advertising Costs and Effects

The analysis stage examines costs and effects associated with various communication channels and advertising media. For instance, in Internet advertising, typical cost considerations are cost per mile (CPM), cost per click (CPC), and conversion rate. These terms have strong connections with mathematics and statistics.

For Web advertising, CPM usually refers to the cost for every thousand visits to the publisher’s Web site. For example, assume that an ad network offers a $5 CPM for a banner, which was put on three Web sites for three months. If the total page views for the three Web sites are 80,000, 110,000, and 140,000 during the three-month period, the total cost of Web advertising for the ad network is

In general, if an ad is posted in n Web sites, the total cost is

where W_i is the number of Web impressions (visits) to the i^th publisher’s Web site for the same period of time.

Consider that the number of Web impressions on each publisher’s Web site depends on many continuously changing factors; then W_i is a random number. Let E(W_i) be the expected value of W_i, which measures the long-term average of the number of Web impressions of the banner on the i^th publisher’s Web site. The long-term average cost is

CPC refers to the amount that the advertiser pays for each click generated from the Web publisher. For example, if the cost per click is $0.04, and three Web publishers generate 1700, 1600, and 900 clicks in three months, the cost of Web advertising is

In general, if a Web ad is posted in m Web sites, the total cost is

where C_i is the number of clicks generated on the i^th publisher’s Web site for a given period of time.

Consider the fact that the number of clicks on each publisher’s Web site depends on various unexpected factors: C_i is actually a random variable. Let E(C₁) the expected value of C_i, which measures the long-term average of the number of clicks generated from the i^th publisher’s Web site over a given period of time. The long-term average ad cost is then

The foregoing two concepts, CPM and CPC, measure the potential impact of the internet ad only in terms of clicks or Web visits. However, these two concepts are unable to provide the advertiser with information regarding whether the Web impression has been transferred into the desired action (such as buying the advertised product). A useful measurement in Web advertising to help account for the advertising effect is the “conversion rate” (or CR, the average number of people taking the action encouraged by the ad per 100 visits to the publisher’s Web site). For example, if out of 2000 clicks on an ad posted on a publisher’s Web site, 12 people end up buying the product, the conversion rate of the ad for this Web site is then

Being highly associated with key factors such as the design of the publisher’s Web site, the conversion rate is an index that directly measures the final impact of the ad for the Web site.

Since the conversion rate directly reflects the performance of the Web site, it can be used to compare advertising effects of two or more Web sites. However, it is risky to compare conversion rates directly. The example in Figure 1 helps illustrate this point. Consider two Web sites: Google AdSense and Chitika. If the conversion rates of the two Web sites are as follows in the past four months, it is impossible to claim which site has better performance on Web advertising.

In fact, the raw values shown in Figure 1 include the stochastic influence of many online factors. In this case, to evaluate the monthly advertising effect of different Web sites accurately, statistical data analysis is needed.

Because of random effects, the expected value of the conversion rate of each Web site should be considered when comparing two or more publishers’ Web sites in terms of the conversion rates. Given a set of historical data involving all the Web sites of interest, one of the statistical estimation approaches is the method of “simultaneous confidence intervals,” which compares the ranges of expected conversion rates with a pre-specified confidence level. For example, with a set of data for the conversion rates of three Web sites over a period of time, if a 95% simultaneous confidence interval reads

and

it means that at 95% confidence level, the advertising performance (in terms of conversion rate) of Google is better than that of Chitika and Yahoo.

To enhance the accuracy of the simultaneous confidence ranges, or to improve the power of testing multiple advertising effects, the two-stage estimation procedure can be considered. When the underlying distribution of the monthly conversion rates is skewed, the two-stage estimation procedure can be used with nonparametric tests to make inferences on the performance of multiple Web sites.

Data Mining and Advertisements

Masses of personal data being collected every day about consumers, via mechanisms like credit card applications, consumer discount cards, and product views and ratings on shopping Web sites are poised to revolutionize the field of advertising. Data mining is the mathematical and statistical method for sifting through large volumes of data to find patterns and create prediction models, in this case of consumer behavior. In 2009, the online video rental company Netflix awarded a $1 million prize to the winners of its three-year contest to develop a better algorithm to predict what movies its users would prefer, based on ratings data provided by the company.

Finally, mathematics is used not only to decide when, where, and how to advertise products and services but also to determine what to emphasize within the advertisements themselves: discounts on pricing or the number of calories per serving, just to name two. However, it is often difficult to verify those numbers. Many will remember Trident Gum’s 1960s slogan, “Four out of five dentists surveyed would recommend sugarless gum to their patients who chew gum.” Although the statement was popular at the time, its legitimacy was later questioned, since it came from a survey whose details have never been released.

IBM has initiated a Smarter Planet campaign focused on dispersed or cloud computing (Internet-based computing). Its “Smarter Math Builds Equations for a Smarter Planet” commercial cites mathematics as the universal language and gives a number of ways in which mathematics will be used to create a “smarter planet.”

Further Reading

Baines, Paul. “A Pie in the Face.” Alternatives Journal 27, no. 2 (2001).

Graydon, Shari. Made You Look—How Advertising Works and Why You Should Know. Toronto: Annick Press, 2003.

Kotabe, Masaki, and Kristiaan Helsen. Global Marketing Management. Hoboken, NJ: Wiley, 2004.

Laermer, Richard, and Mark Simmons. Punk Marketing. New York: HarperCollins, 2007.

Murray, David, Joel Schwartz, and S. Robert Lichter. Ain’t Necessarily So: How the Media Remake Our Picture of Reality. New York: Penguin, 2002.

Russell, J. Thomas, and W. Ronald Lane. Kleppner’s Advertising Procedure. Upper Saddle River, NJ: Prentice Hall, 1999.