[sesqu]

My understanding of the argument is that you have to assume you keep playing after you lose, so your stake is dictated by the risk. For the example in the article, the potential upside is 110% profit and the potential downside is 100% loss, so the optimal stake is relatively conservative in order to prevent any loss from impacting your future ability to invest.

Imagine you get the expected result of one win and one loss and had staked 5%: you'll end up with 1.055×0.95=1.002 wealth. If you had picked a 50% stake, you'd be sitting at 1.55×0.5=0.775 wealth instead, since the loss more than erases your winnings.

The unstated contrast is to models where e.g. there are only a limited number of investment opportunities, losses aren't total, or there are capital infusions.

[n4r9]

Apologies for the confusion but I was talking about Samuelson's argument against the Kelly criterion.

If I was to make one gamble per day and wanted to maximise my profit after 50 years, I don't know if I'd use the Kelly criterion. I need to think about it.

[OscarCunningham]

If you just want to maximise the average amount of money that you end up with then you should bet your entire fortune every day. Of course you will be bankrupt if you ever lose your bet, which is very likely over the course of 50 years. But in the ridiculously unlikely event that you won every bet your fortune would be even more ridiculously large. So the average amount of money would be very large, even though you would very probably be bankrupt.

[n4r9]

I see. What if I wanted to maximise the expected profit, given that the chance of bankruptcy should not exceed 1%? Does that sort of problem become difficult to solve?

[OscarCunningham]

Just always keep a cent in reserve. Then you can never go bankrupt and your average winnings are barely affected.

[n4r9]

That's a good point.

Sorry for the late reply (don't know if you'll see this!) but I happened to be thinking about this again and the question popped into my head: what if I wanted to maximise a specific percentile (e.g. the 5th percentile) of my total winnings?

[sesqu]

Fixing the percentile seems akin to the gambler's fallacy: since losses below a certain point don't matter, you just have to bet everything when you fall off, like you would if you thought after a big loss streak you were "owed" a win.