A public relations agent claims to have come up with a formula which “illustrates” the decline in fame of a brand or celebrity (he treats these two as synonymous) over time.
Mark Borkowski is a regular Guardian columnist and head of Borkowski PR. By happy chance the article in the Guardian where he introduces his formula coincides with the publication of his book today, where he discusses it, and celebrity in general in more detail. I hope the £16.99 price tag is worth it as it’s difficult to make sense of his mathematics just from reading the article.
Mr. Borkowski wants us to know that (a) if you have a brand or are a celebrity your fame will decline dramatically over time, and (b) a good publicist can help prevent this and keep you in the public eye by judicious planting of stories and attention-grabbing vignettes. The trouble is that most celebrities and brand managers know this already and wouldn’t see the need to purchase Mr Borkowski’s services, so it’s a good thing Mr Borkowski has developed a quantitative means of assessing fame which will give him the edge over his competitors. This is his fabulous formula:
Where F is the level of fame, T is time measured in three monthly intervals, B is a baseline of fame calculated from the average level of fame before the peak, and P is the “increment” of fame above the baseline, “that establishes the individual firmly at the front of public consciousness”.
The formula is said to illustrate “that without intervention in the form of further publicity, fame follows an exponential slide to obscurity.” Borkowski goes further than this and states that the slide to obscurity lasts about fifteen months. He shows us this by substituting T=5 into the formula (representing fifteen months) to obtain F = B + 0.04P, showing that the fame-boost received at the height of public attention has been reduced by 96%.
One is tempted to think that the formula is a bit of a gimmick, since publicity and the level of media interest in a brand or celebrity is a hard thing to quantify. Nevertheless, one would expect, given reasonable values for the parameters and the inputs, that the formula would give plausible outputs. However, those who do think this would be reckoning without a shocking ignorance of basic mathematics on the part of Mr Borkowski, and his assumption of the same ignorance on the part of the Great British Guardian reader, for his formula is a load of bollocks. For those who wish for a restoration of mathematical sanity there is the formula that Mr Borkowski should have used at the end of this article. It took about twenty seconds to draw up.
The problem with Mr Borkowski’s formula is that it gives nonsensical values across an important part of the range over which it is supposed to apply. Here’s an example: What fame value does it give at one month after the initial kick of publicity? At this point T = 1/3. If we plug this into the formula we get the answer F = B + 4.8P which is more than 4.8 times higher than the supposed high point is above the baseline.
There is also a problem in that at T = 0 (the starting point) the formula cannot produce a fame value. As the value of T gets smaller and smaller (approaching zero) the amount of fame gets larger and larger, and approaches infinity. Mr Borkowski thinks that at t = 0 the formula gives an infinite fame value, which he acknowledges is not accurate, but nonetheless thinks appropriate as it puts people in mind of the fame value being “off the radar”. This is untrue: the formula does not give a fame value at T = 0 as the operation necessary to calculate it cannot be performed. Mr Borkowski is clearly sacrificing accuracy to appearances here. Further details at the end for those who can stomach it.
Here is a graph showing what Mr. Borkowski wants the formula to do, and the formula he should have used:
And here is graph of the formula he has actually used. Time is in Mr Borkowski’s units of three month intervals, notional parameters of a baseline B of 1 and initial kick P of 1 have been used:
Call me a cynic, Mr Borkowski, but isn’t your formula part of an avaricious (though mathematically inept) scheme to increase sales of your book by fooling people into thinking you’ve placed PR on a scientific footing? An A-level maths student could have told you you’d used the wrong function to model the data because it doesn’t give sensible answers across the time period, but you went ahead anyway and fooled yourself into thinking that the discontinuities gave it extra cachet, rather than being an indication that you’d got something wrong. The irony is that you would have got away with it if you’d only used the formula given above. How much do you pay your mathematics consultant?
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A More Technical Addendum
Let’s look at the formula in more detail. Mr Borkowski does not give us any of the data he obviously used to derive it. He does not tell us any values for the baseline fame B, or the initial increment P given to the celebrity or brand: some sort of example would have been helpful here, because it could have been used to check his claims. But there’s another nagging problem that only sad geeky people would be interested in; you see, despite stating that a celebrity’s fame follows an exponential slide, this formula is a power function and not an exponential one. An exponential function would look like the formula given above and reproduced here:
where the time variable is in the index or exponent (hence the name). The difference is important because if he had used a function like this Mr Borkowski might have earned himself a little more credibility. The problem is that his original formula is what mathematicians call “undefined” at t = 0, and that’s because of the 1/T and 1/T^2 (the ^2 means “to the power of 2″ or “squared”). When you substitute zero into the formula wherever T is you end up trying to divide by zero, which is an operation in mathematics that is not allowed.
To see why, remember that dividing one number by another is equivalent to subtracting the second number repeatedly from the first until you reach zero. So 6 divided by 2 gives an answer of 3 because when you subtract 2 repeatedly from 6 you can do it 3 times until you reach zero. Now imagine dividing 6 by zero. It doesn’t matter how many times you take 0 away from 6, you won’t budge at all from your starting number. Hence you’ll never reach zero. The division process is not working and that’s why the operation is called undefined. Computers have to be specially programmed to recognise when they’re getting into a situation where they’ll be asked to divide by zero, or they’d plunge themselves into an endless loop trying to achieve it.
