The optimal long term strategy for managing a portfolio of independent investments is to always pick a mix that maximizes the expected value of the log of your net worth. This leads to a more conservative investment strategy than the naive "maximize your expected value", and explains such things as why money-losing investments into buying insurance can be a really good idea.
In general this is probably not a bad life strategy.
Let's use the back of the envelope that of VC backed companies, 10% are great successes, 60% die, and 30% will last a good amount of time but don't recoup the investment. The average employee in a successful startup will get a nice payday, but not exactly a life changing amount.
This is doubly true for people capable of being software developers. Your expected income from work is already sufficiently high that a million dollar payday does not change the log of your net worth by that much. Having worked in a hot startup is good for your salary, but usually not by a factor of 2 let alone enough to really change the log of your net worth.
The end result is that it is economically irrational to give up, say, 5% of your salary in return for a chance at hundreds of thousands if the startup sells for hundred's of millions.
Now there are lots and lots of reasons to be an employee at a startup. If you do, there are lots and lots of reasons to pick one that you think has a good shot. But the hope of becoming rich off of options is only one of them if you derive great entertainment value from it.
I get that your utility function from money is non-linear, but I would expect a more accurate model to be a step-function, with large steps at "out of debt", "can tell a bad boss sayonara", "can buy a house", "can pay for kids' college eduaction", and "never have to work again". Equity payouts from a typical startup exit often line up nicely with the middle three, and if you hit the Google/Facebook jackpot, sometimes the last.
Log is often used because it makes linear proportional returns. It makes the most sense when gains are considered derived from the principal, ie proportional returns from capital. See an example below.
As others have mentioned, maximizing log is equivalent to maximizing the underlying.
But when considering returns on accruing capital, a 20% loss is much worse than a 20% gain is good, and similarly a 100% gain is much less good than a 100% loss is bad. This is correctly captured by taking the log. In the case where money is simply accrued from some external source, and their is no proportional return, log isn't necessary.
Independent investment opportunities generally have the effect of multiplying your value by a random amount over a specified interval. When you take logs, you are adding a random amount instead. From the strong law of large numbers, after enough intervals, it is statistically certain that the sum of the logs of those random numbers numbers will converge on the number of intervals times the expected value of the log of your investment strategy.
Therefore maximizing the expected value of the log maximizes the long term rate of return that you (with 100% probability) will observe.
Well, he's suggesting maximizing the expected value, which is a little different. For example, a situation where you have a 10% chance of making $10M and a 90% chance of making zero has an expected value of $1M, but its expected value in log10 space is 0.7 = only about $5, while a 100% chance of making $100K has an expected value of $100K, but in log10 space = 5 = $100K.
I still quibble about the use of log for this purpose (and even if you do, what base?), but I see his point. A non-linear utility function penalizes low-chance but high-value outcomes.
The most general (correct) form of this statement is "use the expectimax algorithm". Yeah, it's a bit weird that he's prescribing log as the mapping from dollars to utility.
If log is the correct mapping for you, it actually doesn't matter what base you use. It just comes out as a constant factor on the expected utility.
How does the expected value of the log of your net worth deal with the possibility of a negative net worth? Any finite probability of zero net worth will weigh infinitely in the log domain, right?
The optimal long term strategy for managing a portfolio of independent investments is to always pick a mix that maximizes the expected value of the log of your net worth. This leads to a more conservative investment strategy than the naive "maximize your expected value", and explains such things as why money-losing investments into buying insurance can be a really good idea.
In general this is probably not a bad life strategy.
Let's use the back of the envelope that of VC backed companies, 10% are great successes, 60% die, and 30% will last a good amount of time but don't recoup the investment. The average employee in a successful startup will get a nice payday, but not exactly a life changing amount.
This is doubly true for people capable of being software developers. Your expected income from work is already sufficiently high that a million dollar payday does not change the log of your net worth by that much. Having worked in a hot startup is good for your salary, but usually not by a factor of 2 let alone enough to really change the log of your net worth.
The end result is that it is economically irrational to give up, say, 5% of your salary in return for a chance at hundreds of thousands if the startup sells for hundred's of millions.
Now there are lots and lots of reasons to be an employee at a startup. If you do, there are lots and lots of reasons to pick one that you think has a good shot. But the hope of becoming rich off of options is only one of them if you derive great entertainment value from it.