Well if we’re only talking about long-term:
In my view the optimal amount of derailing for the user is none or low, whereas the amount of derailing that maximizes revenue for beeminder is somewhat higher.
Too much derailing means your pledge is too low or your rate is too high. But more derailing means more revenue for beeminder as long as it’s sustainable.
- There’s an exception: If the pledge is very low, more frequent derailing might actually mean less revenue than less frequent derailing at a higher pledge, but this isn’t guaranteed - it depends on how the user’s motivation increases with money. If there’s a sharp cutoff at the motivation point (say $100), user awesomeness is maximized when the pledge is just above the motivation point ($101), whereas revenue for beeminder is maximized when the pledge is either insanely high ($10,000) or sustainably below the motivation point ($45). $75 might not be sustainable, and $25 might be too low revenue.)
In my view, as derailing increases, once you’re past the point when you’re choosing to derail once in a while (beenice), awesomeness for the user starts decreasing, yet (long-term) beeminder revenue continues to increase - up until we get to derailing levels so high that they are not sustainable because the user quits out of frustration.
No/little derailing - could mean either high awesomeness because you’re at your motivation point, or medium awesomeness because your rate is below optimal.
— Corresponds to medium income for beeminder long-term.
Medium derailing - could mean derailing is by choice, but then your rate is probably too high, so low awesomeness. could mean your pledge is below optimal, so low awesomeness. Users will probably not get too frustrated, though, so this is probably sustainable. Could mean it’s not the right goal for you.
— Corresponds to high income for beeminder long-term, if sustainable.
Lots of derailing - probably means beeminder is not working for you. Pledge may be below optimal, rate may be too high, or not the right goal for you. Likely to not be sustainable and lead to frustration.
— Corresponds to low income for beeminder long-term because it isn’t sustainable.
Each user has a level and type of akrasia corresponding to a function of two variables - frequency of keeping your pledge is a function of the pledge and rate.
Beeminder revenue is pledge times frequency of not keeping your pledge.
For each user and goal there is an ideal rate i. The rate the user actually performs will be the rate they set times the proportion of times they don’t derail, or rf. The user wants this to be as close to i as possible, that is, to minimize |rf - i|. The user also wants to minimize payment to beeminder, that is, minimize p-pf.
So we can model awesomeness to the user as a(p-pf) + b(|rf - i|). The maximum awesomeness is 0. a is a negative constant corresponding to how much it sucks to pay each dollar to beeminder, and b is a negative constant corresponding to how much it sucks to do the task more or less than the ideal rate.
@dreev’s theory that revenue is proportional to awesomeness can be expressed as c(p-pf) = a(p-pf) + b(|rf-i|).
This can be rewritten as:
(1) n(p-pf) = |rf-i|
where n is the new constant (c-a)/b.
(1) essentially is a claim about akrasia - that akrasia works in a way such that users’ akrasia functions generally satisfy, or come close to satisfying, (1).