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Interesting. I don't know much about economics or the Kelly criterion, but my understanding of the argument is that maximising the growth rate is distinct from
by n4r9 8y ago
Interesting. I don't know much about economics or the Kelly criterion, but my understanding of the argument is that maximising the growth rate is distinct from maximising actual profit arbitrarily far into the future.
Would be cool if there was a simple toy example demonstrating a situation where these differ. Perhaps it even differs for the example in OP.
- norswap 8y agoHow is the growth rate determined? Since presumably it varies, is it averaged in some ways?
- imh 8y agoIf a big growth rate is a composition of a bunch of small changes with their own growth rate, then the overall change is the product of the individual ones, so geometric mean is appropriate.
- n4r9 8y agoI wrote a reply, but accidentally made it top-level: It's the expected growth rate, so yes it would vary in a real-world instance. I wondered for a while why they've taken the logarithm, but I think it's just because growth models are normally defined exponentially (with the rate being a parameter inside the exponential) - it shouldn't make a difference to the result. As for varying, that Samuelson article says: > For N as large as one likes, your growth rate can well (and at times must) turn out to be less than mine - and turn out so much less that my tastes for risk will force me to shun your mode of play.
- sesqu 8y agoMy 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 8y agoApologies 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 8y agoIf 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 8y agoI 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 8y agoJust always keep a cent in reserve. Then you can never go bankrupt and your average winnings are barely affected.