Would you prefer a 50% chance of gaining €10, one chance in a million off gaining €5 million, or a guaranteed €5? The standard position on Less Wrong is that the answer depends solely on the difference between cash and utility. If your utility scales less-than-linearly with money, you are risk averse and should choose the last option; if it scales more-than-linearly, you are risk-loving and should choose the second one. If we replaced €’s with utils in the example above, then it would simply be irrational to prefer one option over the others.

There are mathematical proofs of that result, but there are also strong intuitive arguments for it. What’s the best way of seeing this? Imagine that X₁ and X₂ were two probability distributions, with mean u₁ and u₂ and variances v₁ and v₂. If the two distributions are independent, then the sum X₁ + X₂ has mean u₁ + u₂, and variance v₁ + v₂.

Now if we multiply the returns of any distribution by a constant r, the mean scales by r and variance scales by r2. Consequently if we have n probability distributions X₁, X₂, ... , X_n representing n equally expensive investments, the expected average return is (Σⁿ_i=1 u_i)/n, while the variance of this average is (Σⁿ_i=1 v_i)/n². If the v_n are bounded, then once we make n large enough, that variance must tend to zero. So if you have many investments, your averaged actual returns will be, with high probability, very close to your expected returns.