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Comment by btilly | Hacker News Reader

I'm sorry, but you are plain wrong. The log has nothing to do with utility. And there is no chance of really understanding the result if you're confusing yourself with that bad idea.

To start, EVERY utility function that is both increasing and sublinear will agree that Kelly is the best strategy. Whether square root, log, or bounded - it doesn't matter. The details of your utility function are unimportant.

What matters is that each iteration of an investment strategy multiplies your net worth by a random factor. But log turns multiplication into addition. And statistics has very strong results about sums of independent variables.

The result is that with 100% odds, a player following Kelly will eventually wind up ahead of any other static strategy that you could choose. Both wind up ahead and eventually remain ahead. Which is why a wide variety of utility functions will conclude that Kelly is the optimal strategy.

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> To start, EVERY utility function that is both increasing and sublinear will agree that Kelly is the best strategy. Whether square root, log, or bounded - it doesn't matter. The details of your utility function are unimportant.

This is simply false. This is easy to check for the sqrt utility case. You can calculate the optimal proportion for a single bet and note that it's different than for Kelly, and then you can calculate the utility-function-given-that-you're-about-to-make-a-bet and check that it's still proportional to sqrt. So by induction you are always going to bet the same proportion no matter how many bets you have to make, and this proportion is different from Kelly.

> The result is that with 100% odds, a player following Kelly will eventually wind up ahead of any other static strategy that you could choose.

This is true in the sense that the probability tends to 100% as the number of bets tends to infinity. But this doesn't make Kelly optimal, because in the event that the Kelly isn't ahead the expected utility of the other strategy could be much higher than Kelly.

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.

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?

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