The golden mean of Beeminder: If you never derail, dial up your bright red lines — if you’re derailing all the time, dial them down. Thanks to felixm and others in the forum for inspiring this post.
The previous blog post in our series on how derailing is good-actually (see the sidebar for a review) ended thusly:
The (meta) goal is to be pushed to do as much as possible — or whatever maximizes the motivational value you get from Beeminder — at minimal cost.
Shall we do math to that?
Backing up, we’ve been making a gradual shift in Beeminder philosophy from the early days. We used to lean in to the punishment framing, where pledge payments were penalties for falling short of your commitments and quantified how bad of a person you were. Just kidding. But we’re now pretty serious about treating it as the exact opposite of that. The pledges you commit towards your goals are helping you make progress on your goals, even when you pay them.
Sidebar: The “derailing is good-actually” series
- “Bee Nice To Yourself” (2014) planted the seed for the rest of this list. It’s our cofounder, Bee, pointing out that treating oneself to a derailment now and then is ok. It even helps clarify the value you have for various behaviors. Bee asks herself, for example, “how much do I want to not go for a run right now?” If it’s less than $10, she’ll run. If it’s more, she’ll pay. And if she gets up to $90 at stake and still does’t want to go running, she’ll re-evaluate why she’s trying to get herself to run in the first place.
- “Derailing Is Not Failing” (2019) argues that Beeminder revenue is proportiional to user-awesomeness — that pushing yourself hard enough that you sometimes derail is great for us and great for you. You don’t know how much more you could be accomplishing unless you sometimes find your limits!
- “Paying Is Not Punishment” (2022) is a prelude to the subsequent announcement of No-Excuses Mode and advocates for a generally less excuse-making and more results-oriented mentality with Bee minder.
- “Derailing It Is Nailing It” (2023) turns the concept up to eleven and gives it a positive spin.
- “Beeminder As Your Personal Pigouvian Tax” (2024) argues for a reframing from a punitive to a taxing mindset, where you treat Beeminder’s stings as a behavior-shaping tax and accept that some derailments are inevitable. You can view that tax as the cost of the service Beeminder provides: nudges or rumble strips keeping you on track.
- This post (2025) refines the derailing=good advice. The ideal amount to derail on Beeminder is not never and it’s not constantly. You don’t need to solve a calculus problem but do dial in the slope of your bright red line and your pledge cap to find your happy medium by feel.
Why not not maximal motivation at zero cost?
This is theoretically possible! Nick Winter advocates for it in his 2013 post, Spiraling Into Control. And see especially his classic book, The Motivation Hacker. We’re in awe of Nick but we don’t think this is sustainable (for anyone but Nick?). Or at least that it entails non-monetary costs, like stress and elaborate setup. Nick had to literally commit to writing a book about it to make that all work.
The extremes
If we don’t think the best of both worlds is realistic, how do we optimize this? Let’s first consider two extremes.
At one extreme is a total lack of ambition: A flat red line, no risk, no cost, nothing accomplished beyond what you would’ve accomplished with no Beeminder graph at all.
At the other extreme is maximal ambition: your red line is as steep as is humanly achievable, your pledge is at the edge of affordability, and so you get the maximum accomplished at high cost.
Neither of those sound great and they suggest a happy medium in which you accomplish a lot and derail sometimes but not too often.
So the final section of this post is just for fun. We don’t actually have to do math to this. In reality, you can and should dial in your sweet spot by feel. If you haven’t derailed a goal for months, make it a bit harder. If you’re derailing every week, ask yourself: is it that the goal is too hard, or is it too cheap to be motivating? Adjust the stakes or the steepness of the bright red line accordingly. If it’s stressing you out, drop the pledge and/or dial down the commitment. Maybe keep reassessing week by week.
The calculus problem
Ok, but pretend you’re a robot, or homo economicus, and can model everything about your life with a nice clean equation. (Even I wouldn’t actually do this; like I said, this is just for fun.) We’ll assume that:
- \(s\) is your stakes
- \(r\) is the daily rate of your bright red line — the amount of work you’ve committed to do per day
- \(R\) is the maximum possible value for \(r\)
- \(v\) is the value you get per unit of work accomplished
- \(u\) is the probability of an unavoidable derailment (we’re assuming No-Excuses Mode here)
- \(\lambda\) and \(\alpha\) are tunable parameters determining how steeply your risk of derailing drops as your stakes (\(s\)) go up and how much it increases with your rate (\(r\)).
So we’re treating \(v\cdot r\) as your total daily motivation, assuming you don’t derail. Next, assume your daily probability of avoidably derailing is (why not?) like so:
$$ d(r,s) = \left(\frac{r}{R}\right)^{\alpha} e^{-\lambda s}. $$
So your total probability of derailing is \(d(r,s)+u\). Now we can work out your expected motivation per dollar:
$$\text{MP\$}(r,s)=\frac{\text{motivation gained}}{\text{dollars lost}}.$$
Which is:
$$ \frac{(1-d-u)\cdot v\cdot r}{(d+u)\cdot s}. $$
Inserting \(d(r,s)\) and simplifying yields:
$$ -\frac{r v\left(\left(\frac{r}{R}\right)^\alpha+(u-1) e^{\lambda s}\right)}{s\left(\left(\frac{r}{R}\right)^\alpha+u e^{\lambda s}\right)}. $$
That’s strictly concave in \(r\) if \(0<\alpha<1\), so a unique interior maximum exists. Finally we’ve found our calculus problem! For given stakes, the rate that maximizes motivation per dollar is the one satisyfing
$$ \frac{\partial}{\partial r}\text{MP\$}(r,s)=0. $$
Differentiation, whee:
$$ -\frac{v \left((\alpha+2 u-1) e^{\lambda s}\left(\frac{r}{R}\right)^\alpha+\left(\frac{r}{R}\right)^{2\alpha}+(u-1)u e^{2 \lambda s}\right)}{s\left(\left(\frac{r}{R}\right)^\alpha+u e^{\lambda s}\right)^2} = 0. $$
And algebra, whee:
$$ r^{\ast}(s) = 2^{-1/\alpha} R \left(-e^{\lambda s}\left(\pm \sqrt{4 \alpha u+(\alpha-1)^2}+\alpha+2u-1\right)\right)^{\frac{1}{\alpha}}. $$
Here’s how it looks in Mathematica:
So there you have it, gruelingly Goldilocksed. Under a blizzard of contrived assumptions, you now know that when your stakes are at $90 you should dial your bright red line to 29.9% of the maximum you can do, in order to maximize motivation per dollar spent on derailments.