The Happiness Maths

We know that momentary happiness is some kind of function of experience, partially with respect to how that experience compares to previous experiences. We also know that people have hedonic baselines – a basic level of happiness to which they return, irrespective of changes in their quality of life. People win the lottery, and – obviously – they’re delighted. And then in a few months there as happy or as miserable as they ever were. Or they lose their legs, and – obviously – they’re devestated. And then they adjust and end up as happy or as miserable as they ever were.

So, here’s a simple model that explains that phenomena, and maybe does some other interesting things as well:

Momentary Happiness is defined by the difference between your current experience and an average of your previous experiences (with more recent previous experiences weighted more heavily in the average)

The rest of this post is dedicated to exploring a mathematical formulation of this model, and seeing what it implies, what it misses out and how it could be improved. There’s also one eye on the question “How can experience be best manipulated to produce the maximum total happiness?”. If you are not interested in fun with maths, or the role of formal models in aiding thinking, then you might want to give up here.

What’s the point in playing with a model that is so obviously wrong? There’s a million factors it doesn’t include, a myriad ways in which the concepts it tries to formalise are unreal, multiple, contradictory and simply undefined. This, I say, is a Good Thing. Models are abstracts – attempts to see how much explantory or exploratory work we can get done. Simple models are easier to understand than models which include all possible factors (which is a hopeless task anyway – how can you possibly know in advance what all the relevant factors are?). A good model is an unrealistically simple one – it’s like using a map to understand complex terrain. As long as you don’t mistake the map for the territory there’s no harm in using the map find your way.

So, what’s the model? How about

Happiness(t) = Experience(t) – Comparitor(t)

Where happiness(t) is the momentary happiness at any point in time, t, experience(t) is the momentary quality of experience where positive numbers are good experiences and negative numbers are bad experiences and the bigger the more extreme, and comparitor(t) is the weighted sum of previous experiences against which the momentary experience is compared. Let’s define this so that the most recent experiences affect the average more, and there is a limit on how far back in the past the comparitor reaches (so that very old experiences essentially don’t count). You can see precisely how I’m defining this here where I have put the matlab code for the model i’m about to show you. The precise details don’t matter too much though. Here’s some graphs of the basic behaviour of any system defined like this:

Firstly, the hedonic baseline phenomenon:


Here we can see experience (the black dots) improves suddenly (like a lottery win, maybe) and happiness (the blue line) leaps to follow it, but then falls back down to the baseline (which i’ve set at zero – but it could be any constant value, it’s the dynamics i’m interested in). To understand why this happens, look at the comparitor (the red dashed line). At time 15 experience leaps, but the comparitor is still low, so happiness (defined by the difference between them) is high. As experience stays high the comparitor takes account of this higher experience, rising until it matches the new level of experience (ie all points over the time window of the averaging are at the new level). Happiness drops. The system has adjusted, come to expect the new, higher, level of input

(Are you enjoying the scandelously anthropomorphic way i’m using language, by the way? Things are helped by me choosing variable labels like ‘happiness’, which are the same as everyday concepts. You’ll have to trust me when I claim i’m not getting carried away into believing the model to be the truth…)

Here’s what happens when experiences improves and then drops:


Experience improves, and happiness is higher for a bit, but then drops to baseline (as discussed above). Then, when experience drops back to the old level, happiness drops for a bit (before, again, returning to baseline). Why? Because happiness is simply the relative change in experience. It doesn’t matter that things are only going back to how they were before, the system has adjusted to the new level of good experiences (look at the comparitor) and is made miserable by the change down. Bummer.

So the important result from this is that if experience will always return to the baselines, there is no arrangement of experiences which can result in greater happiness. The total happiness at the end is defined by the final level of experience, because every increase in happiness due to improvements in experience is cancelled out by a loss of happiness when experience goes down again.

In fact the only way to be constantly happy is for things to get continuously better:


If you can’t control experience in this way (ie you have a fixed amount of good experience) then the way it is arranged won’t make any difference to the total happiness at the end.

But just because you can’t improve total happiness, doesn’t mean you can’t have peaks of happiness (although they must be paid for with peaks of misery), or even arrange experience so that the peaks are at different heights. A flat experience line provides the same total happiness as an up followed by a down, but the up/down (like the second figure) means happiness reaches a peak value of 1 (which is again, a totally arbitrary value).

Furthermore, if you arrange a negative experience before a positive experience, you get an even higher peak happiness (twice as much, up to 2!):


Because the first bad experience causes the comparitor to adjust to a low level, the same positive experience as before now causes a massive leap in happiness. The total happiness is the same (notice the drop when the positive experience ends), but the arrangement of unpleasent experience before positive makes the peak happiness derived greater.

This is why we go camping, i suspect.

A couple of other results from this model:

1. Even if experience is entirely positive, but to a random amount, there are still peaks and troughs of happiness.


Because happiness is defined by the change in experience, you see. Although all the experiences are ‘positive’, there will be a time when a few really positive experiences stack up (raising the level of the comparitor) and are followed by a good, but not as good, experience. The difference means that happiness sinks really low. Good experiences don’t let you avoid misery

2. If things get worse gradually, then the point of greatest misery is before they are at their worst.


Here i’ve made experience a sinusoidal function of time. You can see that, around point 20, when things are getting bad, happiness is lowest before they are at their worse (and the inverse holds for things getting better). The reason for this is that the point of greatest unhappiness is when things are getting worst fastest, and the system has had least time to get used to things being bad (it can still ‘remember’ the good times). By the time experience is at an absolute minimum the system has forgotten the good times and started to get used to the bad times (sorry, the window of the time averaging of experience doesn’t include the range over which experience was positive, and the more recent large negative experience values are those most highly weighted).

So there’s the basic behaviour of the system. General questions to ask about a model, at this point, are now, I suggest:

  • does it fit the data we started out with? [validation]
  • does it suggest anything interesting to look for? [prediction]
  • if it doesn’t fit all the data, can it be improved? [development]

  • Since all i started with was the idea of hedonic baselines and a desire to explore a simple model, it’d say that the model does what it was intended. The prediction that total happiness can’t be improved seems false, but then maybe some buddhists would disagree.

    Now, re: development, i’m wondering what direction to take the model next:

  • Look at the effect of adding asymmetry (eg things getting worse makes you more miserable than things getting better makes you happy)
  • Change the function that gives the comparitor – perhaps by making it so that you don’t just adjust to experience, but you also adjust to the rate experience is changing at (this would mean that a steady improvement in experience, like the 3rd figure, wouldn’t keep you happy).
  • Hell, you could even make it so the system adjusts to all the different orders of change (ie adjusts to changes in experience, the rate of change, the rate of rate of change, etc) but at different time scales
  • Make happiness partly a function of previous happiness. This feedback would make the model nonlinear and unstable, i suspect (so more interesting?).
  • Any suggestions, anyone?

  • endnotes
    – This, my systems-engineering friends tell me (thanks mike! thanks sean!), is an instance of a ‘finite impulse response model’
    – I kind of imagine that experience is on a log scale to begin with. So an experience with twice the value of another in this model is actually many many times as good (because it causes you to be twice as happy, albeit momentarily, and we know that two cups of tea are not twice as nice as one cup of tea).
    – I think maybe happiness, in this system, is sort of like mood and there is another kind of something, maybe ‘deep satisfaction’, that is defined more by the number of peaks than the simple sum of values over time
    – The evidence for hedonic baselines is good, but people prove suprisingly bad at predicting how life changes will affect their future mood. Daniel Gilbert about this on
    – The basic idea was Nicol’s, in a supermarket in Vauxhall sometime last year

    3 replies on “The Happiness Maths”

    If happiness(t) becomes a weighted average of previous happinesses, i.e. h(t)=a*h(t-1)+b*h(t-2), as well the rest of the model above, then it would still be linear and its stability would depend on a and b. This introduction would allow the model to oscillate which seems to make an intuitive sense when describing mood. My good moods, above my hedonic baseline, perpetuate themselves despite negative experiences.

    The small dent in parsimony would make a large explanatory difference (which is what information criteria are about, i think..?)

    Very interesting stuff Tom.

    Peoples lifetimes are finite, say length N. Society and technology can only afford to provide a certain maximum amout of experience E over that time. How should one partition the experience E over a persons lifetime N?

    It will depend on the duration the comparitor integrates experience over. If we assume that it is less than N then the third graph suggests to me that the allotment of E that one receives should slowly increase throughout life. However, I haven’t bothered yet to do the maths, so I may be wrong.

    Assuming that I am not wrong, then this model suggests that for a happy life we should give people a sparten childhood with many struggles, a difficult youth, a fair middle age, and shower the old with blessings. An curious hypothesis, and counter to many current beliefs, which is a sign to me of an interesting model.

    I was thinking Nicol is right – if we have a limited amount of E we should increase the rate at which we dispense it. But looking back at what Tom has said, one of the main points is that “The total happiness at the end is defined by the final level of experience” or “the way it [E] is arranged won’t make any difference to the total happiness at the end”.

    Society could dispense E at an increasing rate if we aimed for people to always be happy, albeit not at a high level (in graph 3 happiness doesn’t even make it as high as 0.1). However, if society gives a lot of E to people while they are children, they are actually more likely to end up capable of generating additional E for themselves and for the rest of society.

    I can’t remember the statistic exactly, but it’s *something* like: For the average person, 80% of the total NHS lifetime spend is spent during the final two weeks of their life. (Source: some doctor friends of mine, who I may have misquoted!)

    Back to the model. I like that it explains why we go camping. The main flaw is that there’s no way of increasing the total happiness over a lifetime. What I’d be most interested in, is answering the question ‘How much/ how often should I go camping?’. This is just a restatement of Tom’s question which we were keeping one eye on: “How can experience be best manipulated to produce the maximum total happiness?”

    I think the development that should be made to the model is to also adjust to the rate experience is changing. In fact, thinking back to the supermarket in Vauxhall, this is exactly what Nicol suggested originally. (That it should be a first order differential equation.) The solution (max happiness over lifetime) to which was to sinusoidally vary the experience. Does this work?

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