The Cost of Doing Business

In many situations, we need to compare the cost effectiveness of actions. For me, it is spending the limited pot of money on various ways of making one of my books more attractive or more obvious to readers. And for some of these actions, they are all or nothing: the book only has one cover, so I either pay or don’t. But, some can be done over and over: I can advertise a book and then advertise again. Gut instinct suggests doing something repeatedly would be better than once. But is it true? And how might you measure the impact in advance?

Seth Godin blogged today on expected value: the value of an opportunity taking into account the probability of the outcomes.

And I agree totally that the cost/value of a single opportunity is:

Cost/Value~of~Outcome\times Probability~of~Outcome

However, the situation is more complex for repeated opportunities. While his example of a $5.00 cost for parking once is correct, the cost of parking twice is not $10.00.

While the cost/value of the outcome and the chance it occurs remain the same for each opportunity, the expected value of a string of opportunities must take into account the possibility it will happen on some and not others: for example, parking twice could result in no fine, a fine on the first, a fine on the second, or two fines.

So the expected value of parking twice would be:

The expected value that the fine occurs on the first time but not the second:

(^1/_{10}\times^9/_{10}~i.e.~^9/_{100})\times \$50 = \$4.50

Plus the expected value that the fine occurs on the second time but not the first:

(^9/_{10}\times^1/_{10}~i.e.~^9/_{100})\times \$50 = \$4.50

Plus the expected value that the fine occurs on both:

(^1/_{10}\times^1/_{10}~i.e.~^9/_{100})\times \$50 = \$0.50

However, the chances of it happening at least once are also the opposite of it happening at all, therefore:

\$50-((^9/_{10}\times^9/_{10})\times \$50~i.e.~\$40.50)= \$9.50

Either way, for a total opportunity cost of $9.50 rather than $10.00.

So, doing something repeatedly does result in a higher overall expected value, but the mean average expected value per iteration is lower.

Which makes repetition good for things that involve cost and bad for things that add value. In my original example of advertising a book, each time I buy advertising the cost per return is likely to drop.

Of course, as Seth says, working out the probability of an outcome is separate (and potentially complex) question. Especially for non-tangible returns, such as the possibility an advert for a book will make my name slightly more recognisable to a reader when they see another of my books, increasing the chances of a sale of something.

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