Your comment rings my "math applied incorrectly" alarm -- I may just be misunderstanding, e.g. you might be motivated by a logarithmic utility function in amount of money made, but that's a very different thing from the reason we would expect the financial system to be dominated by Kelly-following-financial-entities -- so just in case, let me try to explain my understanding of why Kelly is so important, and why it doesn't obviously seem to be related to the question of whether to start a startup. Any corrections very much appreciated!

Kelly and financial markets

Suppose three investment funds are created in the same year. Let's say the first fund is badly managed and loses 5% of its capital each year; the second fund gains 5% each year; and the third fund gains 10% each year. After 100 years of this, which of the three will be the most important force in the market? I didn't specify that they had the same starting capital, but the first fund is down to 0.6% of its start capital, the second fund has increased its capital 130-fold, and the third fund has increased its by a factor of 13,800, so if they didn't differ by too many orders of magnitude when they started out, the third one beats the others hands-down.

Of course, the growth isn't really constant in each year. Let's suppose your capital grows by a factor of r(i) in year i. Then after 100 years it's grown by a factor of r(1) * r(2) * ... * r(100), obviously, and we're interested in the fund whose strategy maximizes this number, because after a long enough time, that fund will be the only one left standing. We can write this as

r(1) * r(2) * ... * r(100) = exp(log(r(1)) + log(r(2)) + ... + log(r(100)))

and maximizing this number happens to be equivalent to maximizing

(1/100)(log(r(1)) + log(r(2)) + ... + log(r(100))),

i.e., maximizing the mean of the log growth factor.

Now, imagine that the growth your strategy achieves in a particular year doesn't depend on the amount of money you have available in that year: if you have $1 million, you'll buy N shares of ACME Corp, if you have $10 million, you'll just buy 10*N shares instead. Also assume (much less plausibly -- but I'm pretty sure this can be generalized with more difficult math) that the same bets are offered each year, and what happens in one year is statistically independent of what happens in any other year. Then the log growth factors log(r(i)) are independent random variables with the same distribution, so the Law of Large Numbers says that

(1/100)(log(r(1)) + log(r(2)) + ... + log(r(100)))

is approximately equal to the expected value of log(r(i)). Thus, after a long time, we expect those funds to dominate the market whose strategy maximizes the expectation of the log of the factor by which they increase their capital in a given year.

From this, you can derive the Kelly criterion by calculus. You can also see that it's the same criterion as if you only play for a single year, and value the money you have after that year with a logarithmic utility function.

So what about startups?

An important assumption above was that the same bets are available to you each year no matter how much money you happen to have that year. If each year there's a chance that you'll lose all your money, that would be terrible, of course, because it'll happen eventually, and then you are out of the game forever; but barring that, your strategy looks pretty much the same, whether you have $1M or $100M. But if you invest $100K-equivalent in sweat equity in a startup and cash out $10M, you do not tend to re-invest that return by creating a hundred similar startups the next year.

Conversely, suppose your startup fails, and according to some sort of accounting you can be said to have lost 30% of your bankroll in the process. For the above reasoning to apply, not only would you have to start another startup after this (reasonable assumption), but the returns of this next startup, if it succeeds, should be only 70% of the returns your first startup would have yielded -- because see, our assumption was that the return on a successful bet is a constant times the amount of money you've bet (dividends on 10*N shares vs. dividends on N shares), and you've lost 30% of your bankroll, so now you can only be betting 70% of the resources you were betting before.

It seems to me that this makes basically no sense. If you start another startup right after the first one, you've gained experience, you've gained contacts, and it seems that if anything, you should be able to build a better startup this time. Even if not, it seems strange to say that if in some sense you bet 30% of your personal resources in your first startup, then this should imply that your next startup will be exactly 30% worse than the one before, and the one after that will be worse by exactly 30% again. (And that's not even taking into account that you probably won't start enough startups for the Law of Large Numbers to become relevant.)

In conclusion, it seems to me that if the Kelly criterion applies to startups, it must be for a very different reason than why we'd expect to see Kelly-following-financial entities. (Zvi, who has clearly thought about this more than I have, seems to agree with you that it applies in some way, though.) Did that make sense, or did I misunderstand you somehow?