it’s not intuitive to me when it’s reasonable to apply geometric rationality in an arbitrary context.
e.g. if i offered you a coin flip where i give you $0.01 with p=50%, and $100 with q=50%, i get G = √.01√100 = $1, which like, obviously you would go bankrupt really fast valuing things this way.
in kelly logic, i’m instead supposed to take the geometric average of my entire wealth in each scenario, so if i start with $1000, I’m supposed to take √1000.01√1100 = $1048.81, which does the nice, intuitive thing of penalizing me a little vs. linear expectation for the added volatility.
but… what’s the actual rule for knowing the first approach is wrong?
Another way of looking at this question: Arithmetic rationality is shift invariant, so you don’t have to know your total balance to calculate expected values of bets. Whereas for geometric rationality, you need to know where the zero point is, since it’s not shift invariant.
it’s not intuitive to me when it’s reasonable to apply geometric rationality in an arbitrary context.
e.g. if i offered you a coin flip where i give you $0.01 with p=50%, and $100 with q=50%, i get G = √.01√100 = $1, which like, obviously you would go bankrupt really fast valuing things this way.
in kelly logic, i’m instead supposed to take the geometric average of my entire wealth in each scenario, so if i start with $1000, I’m supposed to take √1000.01√1100 = $1048.81, which does the nice, intuitive thing of penalizing me a little vs. linear expectation for the added volatility.
but… what’s the actual rule for knowing the first approach is wrong?
Another way of looking at this question: Arithmetic rationality is shift invariant, so you don’t have to know your total balance to calculate expected values of bets. Whereas for geometric rationality, you need to know where the zero point is, since it’s not shift invariant.
I think the rule is “you maximize your bank account, not the addition to it”. I.e. your value of deals depends on how many you already have.