# Thinking that all individuals pursue "selfish" interest is equivalent to assuming that all random variables have zero covariance

What does that mean, and what does it have to do with random variables and covariance?

Assuming those are the meanings, I don't see what that has to do with random variables and covariance.

My guess: Given an index I, we list all random variables conceivable: ${X_i}_{i \in I}$.

It is clear that $Cov(X_j, X_k) \ \forall j, k \in I$, if well-defined, may or may not be zero.

Is covariance among two random variables an analogy for common interest among two people?

• Yes, it's like saying that the equals sign in mathematics is irrelevant and therefore there would be no "relationships" between entities. Sociological matters and matters of empathy are discussed within the arc of a relationship, not simply as origin nodes with no inter-behavior, so pure selfishness would be pure isolationism, and that contradicts our daily life. Hope that helps.
– sova
Sep 11 '15 at 21:10
• Pursuing selfish interest all the time sounds like giving too much credit for human actions to human will and rationality. Instead, humans are fallible and are easily distracted, impressed upon, misled, and otherwise affected by random and non-random noises. So claiming focused pursuit means claiming that if not all at least these random variables have zero covariance. Sep 12 '15 at 22:57
• There has been a LOT of research done on this and related topics, including game theory on these topics. A good source book with many references is "Liars & Ourliers: Enabling The Trust That Society Needs To Thrive" by Bruce Schneier Sep 16 '15 at 4:46

Taken most literally, the statement says that the claim that people optimize their own well-being without regard for others is equivalent to saying that no random variables ever interact with each other. It seems that the second claim is probably strictly stronger than the first: I can imagine a society in which people optimize strictly for their own well being (for example, a subsistence farming society where families need not interact with other families), but I cannot imagine random variables ever always having zero covariance.

Taken a little more generously - someone making the above claim might be trying to say that the pursuit of one's self interest never benefits or harms someone else. This claim also doesn't seem accurate:

1. There are cases where an individual's self interest hurts someone else, for example, when they are competing for some common thing).
2. There are cases when an individual's self interest benefits someone else, for example when someone is able to specialize in performing some task or providing some product, and trades that product or service to someone else who needs it.

I believe the analogy is meaningful as a critique of the ideal "self" implied in old models of rational self-interest.

Suppose possible "interests" vary randomly within a very large population of selves. To achieve zero correlation among these blind social atoms would still be vanishingly unlikely. Yet this zero correlation is implied in the ideal definition of "selfish."

This is why it is impossible to achieve "perfect diversity" in a financial portfolio. Even without considerations of "human nature," an idealized model of "rational self-interest" is incoherent and worse than useless. In reality, of course, what we mean by "self-interest" is already loaded with correlations.

Let $X$ be any non-constant random variable. Then the covariance between $X$ and $2X$ is not zero. Therefore it is false that all random variables have zero covaraince. Therefore the statement

A. Thinking that all individuals pursue "selfish" interest is equivalent to assuming that all random variables have zero covariance.

is equivalent to the statement

B. Thinking that all individuals pursue "selfish" interest is equivalent to assuming something false.

which in turn is equivalent to

C. It is false that all individuals pursue "selfish" interest.

So either 1) The person who said A intended to express statement C (in a very contorted way for some reason) or 2) the person who said A was confused.