Email is the superior form of communication as I have argued a few times before, but it can sure aggravate your self-control problems. I am here to help you with that.

As you sit in your office working, reading, etc., the random email arrival process is ticking along inside your computer. As time passes it becomes more and more likely that there is email waiting for you and if you can’t resist the temptation you are going to waste a lot of time checking to see what’s in your inbox.  And it’s not just the time spent checking because once you set down your book and start checking you won’t be able to stop yourself from browsing the web a little, checking twitter, auto-googling, maybe even sending out an email which will eventually be replied to thereby sealing your fate for the next round of checking.

One thing you can do is activate your audible email notification so that whenever an email arrives you will be immediately alerted. Now I hear you saying “the problem is my constantly checking email, how in the world am i going to solve that by setting up a system that tells me when email arrives? Without the notification system at least I have some chance of resisting the temptation because I never know for sure that an email is waiting.”

Yes, but it cuts two ways.  When the notification system is activated you are immediately informed when an email arrives and you are correct that such information is going to overwhelm your resistance and you will wind up checking. But, what you get in return is knowing for certain when there is no email waiting for you.

It’s a very interesting tradeoff and one we can precisely characterize with a little mathematics. But before we go into it, I want you to ask yourself a question and note the answer before reading on.  On a typical day if you are deciding whether to check your inbox, suppose that the probability is p that you have new mail. What p is going to get you to get up and check?  We know that you’re going to check if p=1 (indeed that’s what your mailbeep does, it puts you at p=1.) And we know that you are not going to check when p=0.  What I want to know is what is the threshold above which its sufficiently likely that you will check and below which is sufficiently unlikely so you’ll keep on reading?  Important:  I am not asking you what policy you would ideally stick to if you could control your temptation, I am asking you to be honest about your willpower.

Ok, now that you’ve got your answer let’s figure out whether you should use your mailbeep or not.  The first thing to note is that the mail arrival process is a Poisson process:  the probability that an email arrives in a given time interval is a function only of the length of time, and it is determined by the arrival rate parameter r.  If you receive a lot of email you have a large r, if the average time spent between arrivals is longer you have a small r.  In a Poisson process, the elapsed time before the next email arrives is a random variable and it is governed by the exponential distribution.

Let’s think about what will happen if you turn on your mail notifier.  Then whenever there is silence you know for sure there is no email, p=0 and you can comfortably go on working temptation free. This state of affairs is going to continue until the first beep at which point you know for sure you have mail (p=1) and you will check it.  This is a random amount of time, but one way to measure how much time you waste with the notifier on is to ask how much time on average will you be able to remain working before the next time you check.  And the answer to that is the expected duration of the exponential waiting time of the Poisson process.  It has a simple expression:

Expected time between checks with notifier on = \frac{1}{r}

Now let’s analyze your behavior when the notifier is turned off.  Things are very different now.  You are never going to know for sure whether you have mail but as more and more time passes you are going to become increasingly confident that some mail is waiting, and therefore increasingly tempted to check. So, instead of p lingering at 0 for a spell before jumping up to 1 now it’s going to begin at 0 starting from the very last moment you previously checked but then steadily and continuously rise over time converging to, but never actually equaling 1.  The exponential distribution gives the following formula for the probability at time T that a new email has arrived.

Probability that email arrives at or before a given time T = 1 - e^{-rT}

Now I asked you what is the p* above which you cannot resist the temptation to check email.  When you have your notifier turned off and you are sitting there reading, p will be gradually rising up to the point where it exceeds p* and right at that instant you will check.  Unlike with the notification system this is a deterministic length of time, and we can use the above formula to solve for the deterministic time T at which you succumb to temptation.  It’s given by

Time between checks when the notifier is off = \frac{- log (1 - p^*)}{r}

And when we compare the two waiting times we see that, perhaps surprisingly, the comparison does not depend on your arrival rate r (it appears in the numerator of both expressions so it will cancel out when we compare them.) That’s why I didn’t ask you that, it won’t affect my prescription (although if you receive as much email as I do, you have to factor in that the mail beep turns into a Geiger counter and that may or may not be desirable for other reasons.)  All that matters is your p* and by equating the two waiting times we can solve for the crucial cutoff value that determines whether you should use the beeper or not.

The beep increases your productivity iff your p* is smaller than \frac{e-1}{e}

This is about .63 so if your p* is less than .63 meaning that your temptation is so strong that you cannot resist checking any time you think that there is at least a 63% chance there is new mail waiting for you then you should turn on your new mail alert.  If you are less prone to temptation then yes you should silence it. This is life-changing advice and you are welcome.

Now, for the vapor mill and feeling free to profit, we do not content ourselves with these two extreme mechanisms.  We can theorize what the optimal notification system would be.  It’s very counterintuitive to think that you could somehow “trick” yourself into waiting longer for email but in fact even though you are the perfectly-rational-despite-being-highly-prone-to-temptation person that you are, you can.  I give one simple mechanism, and some open questions below the fold.

Given your p*, compute the waiting time without the notifier lets call it T*. Now think of what’s happening during this time interval. You are learning nothing except that time is passing and therefore your p is steadily rising but until we get to the end of this time limit your p is below p*  so you are not tempted. Ok so with that in mind, what you do is reconfigure your beeper to work like this:  if email arrives before T*  your beeper is programmed to  nevertheless remain silent, a grace period if you will, and wait until time T* at which point it finally beeps. And if no email arrives by time T* it starts over again, i.e. it gives you another grace period of length T*.  This continues forever.

By construction this system will beep later than the immediate notification system would have beeped.  So it dominates the stock notifier.  And because we have chosen the grace period of length T*, at every moment of time when you have not yet heard a beep you will assign a probability less than p* that new mail has arrived.  Therefore you will never be tempted and you will only check when you hear the beep.

Open questions:

  1. Is this the optimal mechanism (I think yes.)
  2. Would a randomizing beeper help? (I think no.)
  3. What if your temptation takes a more complicated form:  you are tempted to check when the expected number of waiting emails exceeds a threshold.  What happens then?
  4. State your open questions in the comments.  This is science.