The Principle of Diminishment

Consider the Law of Diminishing Returns, or the Law of Diminishing Marginal Utility in economics. It states that for every additional unit of input added, the amount of output (or utility, whatever) relative to the unit input/cost decreases. The upshot is this: consider two units of input. The output from the two units of input combined will always be less than the sum of the output from each unit of input separately. I guess that may have been confusing to understand, so let I_1 and I_2 be units of input, and let O(I) be the output that results from that input. And so, this paragraph more or less means:

O(I_1 + I_2)<O(I_1) + O(I_2).

Or, generalizing for n inputs,

O(I_1 + ... + I_n)< \sum_{j=1}^{n}I_j.

Now if we think in terms of operators, we can say the “output operator” is nonlinear. That is, contrary to common belief, the gains from hard work do not scale accordingly most of the time. (That doesn’t mean you shouldn’t work hard though, of course.) But I digress.

A few days ago, I had this epiphany: could this nonlinearity exist in other aspects of my life? The more I thought about it, the more I realized that the answer was yes. Examples to follow:

I’ve been very stressed in the past few days, from grades and other things. And I was thinking, well this is bad and I feel very uncomfortable from this, but this does not feel as bad as I would expect from a sum of the stresses from each factor separately. Sometimes I would worry about a lot of one stress factor but very little of the others; other times I would worry a lot about a couple of the others but not of the original one. Each stress factor kinda interferes with every other stress factor trying to stifle me. So maybe this diminishing sum thing works with stress too? [This was the motivating factor for this post.]

And then consider music. If I try to alternate between two pieces of very good music, I notice that each individual piece does not sound as good as it would’ve if I just listened to that piece alone. That is, the total utility from the two pieces combined is less than the sum of the utilities I would’ve gotten from each individual piece.

Also consider two enjoyable activities – eating and watching TV. One derives less satisfaction doing both at the same time, than if he/she does each individually at separate times. There have been multiple studies that corroborate this.

We can say, then, that there could potentially be a principle of diminishment that pervades nature: additional response will decrease for every additional stimulus added. Now it has been said that nature follows from mathematics, which is quite a blessing for us. So we wonder if there some mathematical formulation to the principle of diminishment, and there is.

Consider two arbitrary vectors in a space, \vec{a} and \vec{b}. There is a theorum, known as the Triangle Inequality, that states:

||\vec{a}+\vec{b}|| \leq ||\vec{a}||+||\vec{b}||,

where the double brackets indicate the magnitude.

We can see that equality only occurs when the two vectors are parallel and thus linearly dependent, so for any two linearly independent vectors,

||\vec{a}+\vec{b}|| < ||\vec{a}||+||\vec{b}||,

which is the form more or less taught in HS Geometry. Graphically we can see that the vectors form the sides of a triangle (thus the name):

To cut this short post because I am hungry and I’m failing at writing today, the upshot is: for any two independent, unrelated stimuli, they will tend to interfere with or somehow dampen each other thus effecting this observation. Equality might be reached if the stimuli are “linearly dependent” – if they are directly related to each other (for example I guess, eating and smelling your food).

P.S. this property is called subadditivity, apparently. “Principle of Diminishment” sounds better though.

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