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 =

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 =

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 =

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

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:

- Is this the optimal mechanism (I think yes.)
- Would a randomizing beeper help? (I think no.)
- 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?
- State your open questions in the comments. This is science.

## 14 comments

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March 6, 2012 at 3:18 am

afinetheoremHere is what appears to be an improvement, though certainly not the optimal mechanism:

Randomize in the following way. Let p* be the current belief where I will want to check my email (assume p*>.5, modification is trivial if not). Let T* be the time which elapses, given no information, where I will have p* high enough given the Poisson process. If there is no mail, do not beep at T*. If there is mail, beep with probability

p*+(1-p*)/2=q*

Then, at T*, conditional on no beep, Pr(message arrived| time is T* and no beep) is

(1-q*)/(1-p*+1-q*), which is less than .5. Perform the same randomization again when the posterior hits p* again. Obviously this is sooner than 2T*. But I think it should be straightforward to check that the total number of email checks over any period is lower with this randomizing mechanism.

I imagine the optimal mechanism will do the following: find p*, and randomize such that the when the agent has heard not a beep, his conditional belief is always as close to p* as possible. Any time the agent has a belief that no email has arrived which is less than p*, I should be convoluting that signal with information that the email has arrived.

I note that I saw this post while, halfway through reading an academic paper, I checked my email then browsed the web a little.

March 6, 2012 at 3:24 am

jeffCool. I will give it some thought. Hey what are you doing awake at this hour? Next topic: optimal mechanism for not checking Cheap Talk at 3AM. Answer: get some sleep.

March 6, 2012 at 3:10 pm

Toomas HinnosaarHere is what I think is an optimal mechanism: notifier with delay T*, where T*=-log(1-p*)/r and p* is the threshold probability as before. In this case up to TT* probability is always exactly p*.

By the way, e-mail notifiers already implement Jeff’s mechanism—you can usually choose the frequency of checking.

March 7, 2012 at 8:04 am

AnonymousGreat post as always.

This reminds me a bit of “Communication Can Destroy Common Learning” by Steiner and Stewart.

March 7, 2012 at 1:41 pm

Email management advice « Marc Gawley[…] this article regarding obsessively checking email by the smart professors at Cheap Talk, which uses a Poisson model of email arrival to argue that it […]

March 8, 2012 at 6:17 am

leadership2dayInteresting…and what it reminds me of is simply for most checking email (the behavior) is positive reinforcement – of course, that is if we actually received an email which for most of us could be true.

I like what you have contributed…and my experience is that I schedule everything in real minutes — of the 24-hours of every day. To create a new habit consider scheduling when you “check” email. I do this in 15-minute blocks, 4 times a day. During these blocks of time if I can respond to an email received within 90-seconds I do so and if not I schedule the email as a follow-up in my calendar. Since I use Outlook I drag it into a ‘open’ time in my calendar. So during the few seconds I look at the email I have to determine what needs to be done and how long will it take. I call this Living In The Now. And it becomes the solution to Overwhelm.

I focus on managing myself and what I choose to do in real time…and this keeps me from being reactive, and staying present. I get to choose what I do…and this is very powerful and rewarding. No, I’m not perfect…this is a muscle to be developed and I too have to constantly work at it.

As for willpower — I think this is thought and not a concept based on reality. Put another way it’s an ‘excuse’ for not being clear about the antecedent – what will spur us into action. This is entirely another matter so will no elaborate here.

Again, you have made a great contribution with your post – thank you!

March 8, 2012 at 8:27 am

doncurrieI feel like I am missing something important here. I use Outlook. I have told Outlook to check my mail every 45 minutes (On average a message sits unattended in my inbox 22.5 minutes). I have the mailbeep on. So Outlook interrupts me with mail every 45 but only if there is mail. Of course, I can hit F9 anytime I want and get new mail but why bother?

This process means I can spend 45 minutes heads down working (in my dreams) without interruption (my dream would come true if I could control the knocks on my door, text messages, and phones as well as I can control my inbox). With Outlook’s help this means that important stuff receives a 1-hour turnaround in the worst case and 30-minute turnaround on average. The rest of the stuff was never going to get prompt attention anyway.

Why isn’t this as simple as I think it is?

I wish I could control my door, phones, and text as easily. Solve that problem an you are well on your way to having a great product.

Put another way, my email is the least of my productivity problems. What am I doing wrong?

March 9, 2012 at 11:32 am

Matt GershoffWon’t checking rate be a function of the expected value or expected information content of the email? Seems that expected email value, while possibly amenable to modeling, would fluctuate making an effective notifier difficult to implement.

March 18, 2012 at 7:07 pm

Weekly links for March 18 « God plays dice[…] The Poisson process of e-mail. […]

March 30, 2012 at 7:11 am

Leigh CaldwellThe model is elegant and cute, but the assumption that checking email is a function only of (probability of having received an email) goes unquestioned.

More plausible is that email checking is a function primarily of attention – driven by context and current activity. When I am not especially focused on writing a document, working out a model or writing this blog comment, THAT’s when I check email. Essentially, it’s when I stop doing what I’m doing and my mind is alerted to the possibility of checking email. A beep in this case may well increase the number of times I check.

Admittedly if I learn over time that I only need to look when it’s beeped, my behaviour will change. Paradoxically, this suggests that having the self-control to not look at the email when it does beep might be counterproductive, as it will make me more likely to forget whether I have checked it since the last beep or not.

Of course, this attention-based model is mixed with a probability-driven one: I may also check when I first sit down at the computer, or on other trigger events, a decision which is more likely to be based on time since last checked.

August 6, 2012 at 5:18 am

rikkiprinceReblogged this on Rikki Rants and commented:

Because I just spent an hour looking for this by web searching on Google and Bing, and Twitter searching on snapbird, I thought I should reblog it so it’s somewhere I can find next time!

December 6, 2013 at 4:35 am

Beeps | Jeffrey Ely[…] particular, this paper gives the answer to this question that I first posed in March of 2012. Special thanks to Kevin Bryan and Toomas Hinnosaar for early […]

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