[btilly]

For one iteration? Sure, you can get any answer. However attempting to apply induction to that is wrong because as the number of iterations increases, the range of likely rates of return for each strategy converges, and Kelly is the one that converges to the highest rate.

As for the 100% odds answer, what I said was true is true in the sense that it is actually true. No ands, ifs, or buts. With 100% odds, Kelly eventually wins over any other strategy. Period.

The question of whether this makes Kelly optimal is not the question that the theorem was trying to answer. And therefore is irrelevant. Now in fact this does make Kelly optimal for a wide range of utility functions. But far from all possible ones.

The point being that it is important to separate a mathematical point from our interpretation of what that point implies. When you confuse the two then you get yourself into an unnecessary muddle. Kelly is a statement about the probability of one strategy beating another. It isn't a statement about how you should bet.

[arnioxux]

Using ln as our utility, at time N we get:

  E[ln(X_1*...*X_N)] = sum E[ln(X)]

So kelly happens to maximize this expected utility by construction since it was derived by maximizing expected log of one round of betting (E[ln(X)]).

But if we use square root as our utility instead:

  E[sqrt(X_1*...*X_N)] = prod E[sqrt(X)]

We would maximize this expected utility by maximizing E[sqrt(X)]. Going through the same calculus we can see that we don't arrive at kelly.

Where did I go wrong?