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
where*
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.
I was thinking there should be a constant added which represents the amount of time you are neither happy or suffering …
I really loved your process here. Brought a smile to my face
Here is something of a shorcut to see that the ideal S is at most 1/2. You are optimizing a function which is a quadratic with a negative leading coefficient, so its largest value occurs at the midpoint of the roots. Now one root is in [0,1] (because your function is positive at S=0 and negative at S=1) and the other root is negative (if it were positive, your function would be negative at S=0). So their midpoint lies on the left of 1/2.
I should also mention that the conclusion that at least half of life should be spent
without suffering holds if f(s) is an arbitrary concave function, as you indeed suggest. It is actually not very hard to show the following:
Theorem: If
U(s) = (1-s) f(s) – s,
where f(s) is a continuous, concave, and increasing function on [0,1] which satisfies f(0)=1, then the maximum of U(s) over the interval [0,1] is achieved by some s<1/2.
If that statement sounds interesting to you, I would be happy to sketch out the proof.
I’d love to see that.
Neat. Reminds me of a model I came up with for distributing limited resources, like a loaf of bread among a thousand people. The optimal solution is not to give everyone a crumb which they’ll barely notice, but to give out decent slices until it runs out.
Why is it more realistic for f(S) to be concave rather than convex? Increasing suffering from 1% to 2% doesn’t have as much impact on pleasure as increasing suffering from 98% to 99%.
So let’s set f(S) equal to the ratio of suffering to pleasure: f(S) = S/(1-S)
Then, Total Happiness = [(1-S) * f(S)] – S = 0, so it’s not affected by the amount of suffering.
To make f(0)=1, we can set f(S) = (S+1)/(1-S), in which case Total Happiness = 1, so it’s still not affected by suffering.
Or we could set f(S) = 1/(1-S), in which case Total Happiness = 1-S, so zero suffering is best.
This shows how subjective choices affect the outcome, which is why such models have to be validated.
For an empirical perspective, a recent article in J.Pers.Soc.Psych by Seery, Holman, and Silver finds: “In a multiyear longitudinal study of a national sample, people with a history of some lifetime adversity reported better mental health and well-being outcomes than not only people with a high history of adversity but also than people with no history of adversity.”
https://webfiles.uci.edu/rsilver/Seery,%20Holman,%20&%20Silver%202010%20JPSP.pdf
In other words you took a simple truism {concept} and made it complicated. So much for today’s so called educated mind.
It seems that you’re saying that overcomplication is a poor use of a good mind, and I would certainly agree. However, I think Julia’s post was meant to convey the point that truisms, while simple, are neither very accurate nor very useful, and then she tried to use a rational, mathematical approach to gain an understanding of how it is we really assess the value and happiness in our lives. Does that seem like a better use of time and intelligence?
Ignorance is bliss.
Aw, geez… Did I pet the cat yesterday? Did I enjoy that cookie? Did I get laid recently? Did I enjoy that sunset last night? Did I exercise? Was I nice to people? Did I annoy a miserable crazymaker grump? Now we’re talking happy utility and hell with the math.
You are a graceful web monster of futile unnecessary complications….! Congratulations from Argentinien.
Cocorastuti Von Semberg
Your function is affine, not linear.
Also, for any concave function the optimal S is bounded by 1/2 (oops already pointed out by alex!). In fact for any lipschitz function you can bound it in the same way you already do using the lipschitz constant (for any differentiable nondecreasing concave function this will just be the derivative at 0 in your case).
Finally, I think what you want is F(2*S) = F(S) + 2 — at least that’s usually why one does a log transform to gdp or utility — giving us F(S) = log(S)
Can you explain the motivation for f(2s) = f(s) + 2?
I thought the stars were always the same brightness, but the moon, city lights, clouds and what not made the sky brighter and darker. That’s so disappointing.
Brilliant. I mean, once you ask the question, “How can I mathematically analyze the contribution suffering makes to my happiness?”, the rest follows. But asking the question is brilliant.
I would like to see an analysis done assuming that f is concave, and that Pleasure + Pain + Indifference = 1 (requires another equation to solve). Also, an analysis of happiness and sadness, with the assumption that happiness = d(Pleasure-Pain)/dt = -sadness.
Nitpick: S is a fraction, not a percentage.
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