A Drunkard Walks Through His To-Do List

Photo by St Stev

Sometime in my adult life I acquired a fascination with time management. (This marked part of my transition from slacker status.) The most basic stage of time management is making a to-do list. Over the years I've noticed an interesting phenomenon with this list. I've talked to friends who maintain similar lists and found that their lists show the same behavior, although they never notice it until I point it out and don't necessarily understand it even after I explain it.

This phenomenon is related to what mathematicians (and probabilists) call a drunkard's walk. The name comes from the idea of a drunk attempting to walk home starting from a lamppost. This is not your ordinary everyday drunk, but a mathematically idealized drunk, so that each step he--he because the concept of the drunkard's walk dates back to the dark ages before women were expected to engage in public drunkenness--each step he takes has a completely random direction unrelated to the steps before or after. It turns out you can analyze the drunk's motion in considerable detail, and although it is of course impossible to predict the drunk's exact location at any time (except at the very start), you can make a lot of other predictions, such as that the drunk will return to the lamppost with probabilistic certainty (meaning it is theoretically possible that this would happen, but the probability is zero).

You can model a simple version of the drunkard's walk using coin flips. Suppose he can only choose to go north or south along the street. Start flipping the coin. Every time it comes up heads, move him one step to the north. Every time it comes up tails, move him one step to the south. Probability theory tells us several things about his path, even though his exact position is impossible to predict. Some of these may seem paradoxical:

1. The drunkard's average distance from the starting point (more precisely, the standard deviation) equals the square root of the number of steps taken. For example, after nine steps , the drunkard may be anywhere from zero to nine steps away from the lamppost, but if you take a large number of drunkards each staggering away from his own lamppost, their average distance from the lamppost after nine steps will be very close to three steps. After a hundred total steps the average distance from the lamppost is ten steps, and so on.

2. With probabilistic certainty, the drunkard returns to the lamppost, not once, but infinitely many times.

This is assuming the coin is perfectly fair; that is, the odds of getting heads or tails are exactly the same. If there is even the slightest imbalance--say heads comes up slightly more often than tails, then something quite different happens. The drunkard, although still taking both northward and southward steps, slowly drifts to the north. Pick any spot on the street north of the starting point. The drunkard is likely to cross this position several times, traveling northward the first time, southward the second, and so on, but eventually he crosses the point in the northward direction for the last time, and never comes south of that point again. Given enough time, he travels northward a mile--or a thousand.

Here's what does not happen--in any scenario: the drunkard wanders back and forth within a certain section of street, without going outside it. Even a thousand-mile section of street is not enough room--the drunkard eventually will exit from one end or the other (although he might later go back, depending on the scenario).

Now back to my to-do list. For a long time I assumed that items got added to the list at random. Stuff happens--your car headlight burns out, a raccoon crawls under the porch and dies--and you have to deal with it. And items get subtracted from the list essentially at random, because different tasks require different amounts of time to dispose of. And in the decades that I have been keeping a list, the number of items has always fluctuated between 15 and 50, usually averaging around 30. In particular, I have never completely cleared out the list.

But this doesn't add up, because if the addition and subtraction of list items is random, the length of the list is essentially a drunkard's walk. The drunkard takes a northward step--add an item to the list. The drunkard takes a southward step--subtract an item from the list. The one big difference is that once the list reaches zero, you can't subtract anything else. It's as if there is a wall at the lamppost which keeps the drunkard from traveling further south.

So, two possible scenarios: First is that on average I am able to clear things off the list as quickly as they come in. Because it's random, the list would grow and shrink randomly, but the analysis of the drunkard's walk shows that the drunk would occasionally come back to the lamppost again and again--i.e., the list would shrink to zero sometimes. But my list has never been at zero since I started keeping it.

So the second scenario: Maybe I can't clear items off the list as quickly as they come in. In this case, the analysis shows that the drunkard drifts to the north without limit. In other words, my list would still grow and shrink randomly, but over the long term get longer... and longer... and longer. But that doesn't happen either.

Conclusion? The list is not random. Someone (face it--probably me) is controlling the length of the list. And if I'm not happy with the average length of the list (which I'm not--it's a little long for my taste, although I've decided that zero is not the optimum length) it is within my power to change it.

I've asked friends about their to-do lists, and everyone has the same story. The list fluctuates around some average number of items but never gets too long or too short. Everyone feels controlled by the list but generally doesn't recognize that they must be controlling it.

(BTW the analysis of the drunkard's walk is very robust--pretty much all you need is some randomness either on the addition of items to the list or the subtraction therefrom, and you can draw these same conclusions.)