870: Advertising

Explain xkcd: It's 'cause you're dumb.
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I remember the exact moment in my childhood when I realized, while reading a flyer, that nobody would ever spend money solely to tell me they wanted to give me something for nothing. It's a much more vivid memory than the (related) parental Santa talk.
Title text: I remember the exact moment in my childhood when I realized, while reading a flyer, that nobody would ever spend money solely to tell me they wanted to give me something for nothing. It's a much more vivid memory than the (related) parental Santa talk.


This comic pokes fun at some advertising tricks, analyzing them mathematically.

  • "Up to 15% or more" is a reference to the Geico car insurance commercials: "15 minutes could save you up to 15% or more on car insurance." "Up to" means "less than or equal to," so the phrase means "less than, equal to, or more than 15%," which is a tautology.
The first line uses notation from set theory and reads out as: The union of sets A and B equals the set of all x, such that x is less than or equal to 15, or greater than 15, which equals the set of all real numbers.
Below is a number line (with the numbers being interpreted as percentages) the black dot indicates that the number 15 is included, and the white dot indicates that 15 is not included, but only strictly bigger numbers.
Geico's ad is also referenced in 42: Geico
  • Some things in life are free. However, typically not those aggressively advertised, with a capital "free!" splashed right over the ads, followed by a small asterisk, indicating the presence of a fine print, ensuring that they are only technically not guilty of false advertising. (Get a FREE* drink!)
The little formula Randall gives, is to calculate the least amount of money that they expect to make from you. The suggestion is that they expect their income from the ad to be more than what they paid for it. (*with the purchase of a $6 meal)
  • Some sales are based on a scaling percentage rate - for example, all items are 20% off, but if you spend more than $200, you get 30% off instead, and so on. These are almost universally proclaimed with a phrase like "The more you spend, the more you save!" This is of course nonsense, as "spending" is the opposite of "saving", and the deal is there to make you spend more.
Note that an alternative phrasing "The more you buy, the more you save" is not strictly a contradiction. In some situations, the total number of items you intend to buy may place you just below the spending threshold that would qualify you for an additional discount; you can strategically buy additional items to place you just past the threshold. Despite buying a greater quantity of items, after the discount is applied, you will have spent less money, and thus saved more. (Though savvy retailers will attempt to avert this by placing this exploitable range higher than the cost of what any consumer reasonably needs, forcing them to spend more to qualify for the discount.)

For those who don't celebrate Christmas, Santa Claus is a mythical person responsible for delivering presents to good children on Christmas Eve. Randall explains in the title text that the first flaw in mathematically annoying advertising (that there will always be someone who profits from said advertisement in some way) is what led him to realize that Santa isn't real. Although most small children are traumatized after learning this, it is a testament to Randall's level of geekiness that he is more concerned with the first revelation.


Mathematically Annoying Advertising:
A ∪ B = {x:x ≤ 15 or x > 15} = ℝ
[line graph representing the above equation.]
When discussing real numbers, it is impossible to get more vague than "up to 15% or more".
["FREE!*" in large text, with substantial illegible fine print.]
If someone has paid $x to have the word "free" typeset for you and N other people to read, their expected value for the money that will move from you to them is at least $(x / (N+1))
[Graph representing inverse relationship between "amount you spend" on the y axis and "amount you save" on the x axis.]
It would be difficult for the phrase "the more you spend the more you save" to be more wrong.


Randall changed the image name from advertising.png to mathematically_annoying.png, since adblocking extensions interpreted it as an ad and made the comic blank. He had the same problem again just three months later with 906: Advertising Discovery.

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Am I confused, or is the the third graph wrong with the independent and dependent variables. 21:06, 5 November 2017 (UTC) But the Geico commercial doesn't say up to, it says 15% or more... ~Jfreund

That may depend on your region. 03:24, 30 November 2013 (UTC)
Saying that something "could save you 15% or more" and saying it "could save you up to 15% or more" are the same thing. Both statements take into account the very real possibility that some percentage less than 15 could be saved.Orazor (talk) 13:37, 21 July 2014 (UTC)

Not to mention that Geico says "Could save you..." (In combination with "up to", the "could" should be "will".) Z (talk) 03:09, 18 June 2014 (UTC)

A justification for "The more you buy, the more you save" is that the more discounted products you buy, the more money you save as opposed to buying them at list price. For things we will buy anyway (e.g. food), it may be true. --Troy0 (talk) 20:01, 6 July 2014 (UTC)

Added to the article. --Troy0 (talk) 04:10, 25 July 2014 (UTC)
It doesn't work when the items can expire. Cflare (talk) 14:38, 14 August 2014 (UTC)
it does to a certain point- my family can eat a lot of food before it expires, especially if it's something we like. (talk) (please sign your comments with ~~~~)

Brilliant comic Randall. I wonder what your next one is about.

I used Google News BEFORE it was clickbait (talk) 00:20, 24 January 2015 (UTC)

Doesn't the title text imply that Randall realised nothing is truly free and concluded that Santa wanted something from him, prompting his parents to reveal the big secret? (I conclude this based on Randall claiming that these two events are related) 21:16, 25 April 2015 (UTC)

Given that "up to x or more" must necessarily be true, how can it be "construed as false advertising?" Meaningless advertising, yes; false, no. 04:22, 4 September 2015 (UTC)

I just spent 2% of my life looking for the fine print to that FREE* drink (* given during time of kidney-harvesting scam test. Limit one per customer. No purchase necessary to win. Please see rules to apply.)Beastachu (talk) 10:33, 13 September 2015 (UTC)

Shouldn't the expression in panel 2 be (x+1)/n, not x/(n+1)? If we define Y as how much each person pays, then the company would earn $YN. YN > X ---> YN = X + 1 ---> Y = (X + 1)/N. 03:26, 18 October 2015 (UTC)

Randall defines N as the number of people other than you who read the flier. Therefore the total number of people who got the flier is N+1. The advertiser spent $X to produce the flier and assuming that it wants to make a profit on the advertisement, it needs to make at least X/(N+1) on average for each person that gets the flier. Given this your equation should be $Y(N+1) > X not $YN > X because the total number of people is N+1. Obviously Y(N+1) > X ---> Y > X/(N+1), which is exactly what we already found out. I'm not really clear on how you get the transformation YN > X ---> YN = X + 15:19, 11 November 2015 (UTC)

I've always been "mathematically annoyed" by 'X% off' signs (like "40% off"). OFF from what? From the price they asked for beforehand? But they couldn't sell this particular unit for that price; maybe they didn't even sell any unit at that price (and, even if they did, they clearly got more units to sell than available buyers at that price). So, the X% off is from a meaningless seller-wishful-thinking number, not anything resembling a fair market value (where willing sellers and willing buyers meet). Mountain Hikes (talk) 03:32, 6 January 2016 (UTC)