“How to Count to One Thousand,” J. Sobel (1992)

You have a stack of money, supposedly containing one thousand coins. You want to make sure that count is accurate. However, with probability p, you will make a mistake at every step of the counting, and will know you’ve made the mistake (“five hundred and twelve, five hundred and thirteen, five hundred and….wait, how many was I at?). What is the optimal way to count the coins? And what does this have to do with economics?

The optimal way to count to one thousand turns out to be precisely what intuition tells you. Count a stack of coins, perhaps forty of them, set that stack aside, count another forty, set that aside, and so on, then count at the end to make sure you have twenty-five stacks. If your probability of making a mistake is very high, you may wish only to count ten coins at a time, set them aside, then count ten stacks of ten, setting those superstacks aside, then counting at the end to make sure you have ten stacks of one hundred. The higher the number of coins, and the higher your probability of making a mistake, the more “levels” you will need to build. Proving this is a rather straightforward dynamic programming exercise.

Imagine you’ve hired workers to perform these tasks. If tasks cannot be subdivided, the fastest workers should be assigned to count the first layer of stacks (since they will be repeating the task most often after mistakes are made) and the most accurate are assigned to do the later counts (since they “destroy more value” when a mistake is made, as in Kremer’s O-Ring paper). The counting process will suffer from decreasing returns to scale – the more coins to count, the more value is destroyed on average by a mistake. With optimal subdivision, the number of extra counts needed to make sure the number of stacks is accurate grows slower than the number of coins to be counted, and the optimal stack size is independent of the total number of coins, so counting technology has almost-constant returns to scale.

The basic idea here tells us something about the boundary and optimal organization of a firm, but in a very stylized way. If workers only imperfectly know when mistakes are made, the problem is more difficult, and is not solved by Sobel. If workers definitely do not know when a mistake is made, there still can be gains to subdividing. Sobel mentions a parable about prisoners told by Rubinstein. There are two prisoners who want to coordinate an escape 89 days from now. Both prisoners can see the sun out their window. The odds of one of the two mistaking the day count after that long is quite high, causing a lack of coordination. If both prisoners can also see the moon, though, they need only count three full moons plus five days.

http://www.jstor.org/stable/pdfplus/2234847.pdf?acceptTC=true (JSTOR gated version – I couldn’t find an ungated copy. Prof. Sobel, hire one of your students to put all of your old papers up on your website!)

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