The darker the night, the brighter the stars?
August 24, 2011 18 Comments
“The darker the night, the brighter the stars” always struck me as a bit of empty cliche, the sort of thing you say when you want to console someone, or yourself, and you’re not inclined to look too hard at what you really mean. Not that it’s inherently ridiculous that your periods of pleasure might be sweeter if you have previously tasted pain. That’s quite plausible, I think. What made me roll my eyes was the implication that periods of suffering could actually make you better off, overall. That was the part that seemed like an obvious ex post facto rationalization to me. Surely the utility you gain from appreciating the good times more couldn’t possibly be outweighed by the utility you lose from the suffering itself!
Or could it? I decided to settle the question by modeling the functional relationship between suffering and happiness, making a few basic simplifying assumptions. It should look something roughly like this:
Total Happiness = [(1-S) * f(S)] – S
S = % of life spent in suffering
(1-S) = % of life spent in pleasure
f(S) = some function of S
As you can see, f(S) acts as a multiplier on pleasure, so the amount of time you’ve spent in suffering affects how much happiness you get out of your time spent in pleasure. I didn’t want to assume too much about that function, but I think it’s reasonable to say the following:
- f(S) is positive — more suffering means you get more happiness out of your pleasure
- f(0) = 1, because if you have zero suffering, there’s no multiplier effect (and multiplying your pleasure by 1 leaves it unchanged).
… I also made one more assumption which is probably not as realistic as those two:
- f(S) is linear.**
Under those assumptions, f(S) can be written as:
f(S) = aS + 1
Now we can ask the question: what percent suffering (S) should we pick to maximize our total happiness? The standard way to answer “optimizing” questions like that is to take the derivative of the quantity we’re trying to maximize (in this case, Total Happiness) with respect to the variable we’re trying to choose the value of (in this case, S), and set that derivative to zero. Here, that works out to:
f'(S) – Sf'(S) – f(S) – 1 = 0
And since we’ve worked out that f(S) = aS + 1, we know that f'(S) = a, and we can plug both of those expressions into the equation above:
a – Sa – aS – 1 – 1 = 0
a – 2aS = 2
-2aS = 2 – a
2aS = a -2
S = (a – 2) / 2a
That means that the ideal value of S (i.e., the ideal % of your life spent suffering, in order to maximize your total happiness) is equal to (a – 2)/2a, where a tells you how strongly suffering magnifies your pleasure.
It might seem like this conclusion is unhelpful, since we don’t know what a is. But there is something interesting we can deduce from the result of all our hard work! Check out what happens when a gets really small or really large. As a approaches 0, the ideal S approaches negative infinity – obviously, it’s impossible to spend a negative percentage of your life suffering, but that just means you want as little suffering as possible. Not too surprising, so far; the lower a is, the less benefit you get from suffering, so the less suffering you want.
But here’s the cool part — as a approaches infinity, the ideal S approaches 1/2. That means that you never want to suffer more than half of your life, no matter how much of a multiplier effect you get from suffering – even if an hour of suffering would make your next hour of pleasure insanely wonderful, you still wouldn’t ever want to spend more time suffering than reaping the benefits of that suffering. Or, to put it in more familiar terms: Darker nights may make stars seem brighter, but you still always want your sky to be at least half-filled with stars.
* You’ll also notice I’m making two unrealistic assumptions here:
(1) I’m assuming there are only two possible states, suffering and pleasure, and that you can’t have different degrees of either one – there’s only one level of suffering and one level of pleasure.
(2) I’m ignoring the fact that it matters when the suffering occurs – e.g., if all your suffering occurs at the end of your life, there’s no way it could retroactively make you enjoy your earlier times of pleasure more. It would probably be more realistic to say that whatever the ideal amount of suffering is in your life, you would want to sprinkle it evenly throughout life because your pleasures will be boosted most strongly if you’ve suffered at least a little bit recently.
** Linearity is a decent starting point, and worth investigating, but I suspect it would be more realistic, if much more complicated, to assume that f(S) is concave, i.e., that greater amounts of suffering continue to increase the benefit you get from pleasure, but by smaller and smaller amounts.