Idea Sea

A Simple Model of “Happiness”

(This article was written in 2006)

We know humans never stay at a constant level of happiness from whatever we get. I am curious about how to answer this question: at what level of achievement, we will be able to reach the highest level of happiness.

First, the key variables in this problem are identified as,
H: Happiness
E: Expectation (>= 0)
A: Achievement (>=0)

So I propose the model with its assumptions to analyze the questions raised above.

Assumption I:

An individual’s happiness is proportionate to the gap between his achievement (what he has) and his expectation (what he wants to get), instead of the absolute level of his achievement. In another word, how much you own won’t determine how happy you are. However, how much more you gain than what you have expected will bring you a lot of happiness. Therefore, H (happiness) should be a monotonously increasing function of (A-E). Though I am aware that there are many other ways to describe the relationship, here I tentatively propose a simple linear model as below.
( Eq.1) H = m * (A – E) + n
where m is called an emotion coefficient, and n is an emotion constant. In particular, m > 0, and n > 0. n indicates the happiness level when the person right achieves what he has expected. This model is illustrated in Figure 1 below.

Figure 1 Relationship between Happiness, Achievement and Expectation

Assumption II:

We speculate that people tend to expect more as they achievement rises. Therefore, I propose an exponential relationship between E (expectation) and A (achievement), though there could be other choices for modeling this too. This assumes that, when human achieve something, they would expect more on that base of their achievement. The formula to describe this assumption is as below.
(Eq.2) E = k * exp ( l * A )
where k is called ambition constant, and l is ambition coefficient. In particular, k > 0, and l > 0. It says, more ambitious persons have bigger k and l. k indicates the initial ambition level when his achievement is 0. This is demonstrated in Figure 2 below.

Figure 2. Relationship between Expectation and Achievement

We insert Eq.2 into Eq.1, and then obtain,
( Eq.3) H(A) = m * ( A – ( k * exp( l * A ) ) ) + n
Take the derivatives of H with respect to A, then we get,
(Eq.4) H’(A) = m ( 1 – k * l * exp ( l * A ) )
(Eq.5) H”(A) = – m * k * l^2 * exp ( l * A ) < 0
Because H”(A) < 0, H(A) function is convex, the maximum of H is achieved when H’(A) = 0. According to Eq.4, we get
(Eq.6) A* = – log ( k * l ) / l
Based on Eq.6, we know,
When k*l >= 1, A* <= 0
When k*l < 1, A* > 0
These two situations are illustrated in the Figure 3 below.

Figure 3. Maximum Happiness and Related Achievement
Note: k and l are ambition factors in different aspects.

Therefore, for people with kl >= 1, since A*<=0 and the function is convex, their maximum happiness is reached at the time they haven’t achieved anything (A=0). For people with kl < 1, their maximum happiness is reached at some point of receiving achievements at certain level.

The indication here for people with greater ambition (kl>=1) is that, achieving more makes them less happy. This sounds striking!

This could also be a reminder for such “ambitious” people who want to get bigger happiness via making greater achievement. They’d better stay home doing nothing. Alternatively, they can change their personality first (change their k and l, but can this happen?), and then try making achievements to increase happiness.

Note: due to the simplicity, this model assumes the risk of being misleading…It is just for fun…

Written by Jianxi Luo, September 26th, 2009 at 12:31 pm, Posted in Philosophy

27 Responses to 'A Simple Model of “Happiness”'

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