The darker the night, the brighter the stars?




S = % of life spent in suffering
(1-S) = % of life spent in pleasure
f(S) = some function of S




  •  f(S) is linear.**

f(S) = aS + 1




a – Sa – aS – 1 – 1 = 0
a – 2aS = 2
-2aS = 2 – a
2aS = a -2
S = (a – 2) / 2a




* You’ll also notice I’m making two unrealistic assumptions here:




18 Responses to The darker the night, the brighter the stars?

  1. 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

  2. alex says:

    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.

  3. alex says:

    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.

  4. Max says:

    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.

  5. Max says:

    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.

    • Max says:

      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.

  6. Vesuvium says:

    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.”,%20Holman,%20&%20Silver%202010%20JPSP.pdf

  7. David M. Green says:

    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?

    • Max says:

      Ignorance is bliss.

  8. Ric says:

    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. :)

  9. Cocorastuti Von Semberg says:

    You are a graceful web monster of futile unnecessary complications….! Congratulations from Argentinien.

    Cocorastuti Von Semberg

  10. noah says:

    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)

  11. Kaleberg says:

    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.

  12. Phil Goetz says:

    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.

  13. Pingback: What’s Good Writing? | êtes mischz-vous?

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