Details: Utility is generally considered to be fairly accurately modeled as the logarithm of the total net worth. If the initial net worth is T and the gain from an investment per BTC invested is a random variable X, then the expected gain from investing a is
E[log(T+aX) - log(T)] ~ aE[X]/T - a^2 V[X]/T^2
This is maximized when a = E[X]T/(2V[X]). The approximation requires that E[X] is relatively small, so may not be applicable precisely to the problem discussed.
Interesting.
Do you have a reference where you pulled this from
and/or where I could read more on this topic ?
is a good place to start, as is
http://en.wikipedia.org/wiki/Decision_theory. The usage of a logarithmic utility function is just common sense (it basically means that a person worth $100K is as excited to gain $1K as a person worth $1M is to gain $10K) and was suggested as early as by Daniel Bernoulli; the formula given is just a straightforward calculation (second-order Taylor expansion of log combined with neglecting of E[X]^2, then optimizing the resulting function) (I reproduced the calculation from memory and I had a small error which I now fixed).
Since the approximation only deals with small deviations anyway, changing the utility function doesn't materially affect it, but it does change the effective value of T, which corresponds to the risk-aversity.