The reason I brought this up, which may have seemed nitpicky, is that I think this undercuts your argument for sub-Kelly betting. When people say that variance is bad, they mean that because of diminishing marginal returns, lower variance is better when the mean stays the same. Geometric mean is already the expectation of a function that gets diminishing marginal returns, and when it’s geometric mean that stays fixed, lower variance is better if your marginal returns diminish even more than that. Do they? Perhaps, but it’s not obvious. And if your marginal returns diminish but less than for log, then higher variance is better. I don’t think any of median, mode, or looking at which thing more often gets a higher value are the sorts of things that it makes sense to talk about trading off against lowering variance either. You really want mean for that.
The reason why variance matters is that high variance increases your odds of going broke. In reality, gamblers don’t simply get to reinvest all of their money. They have to take money out for expenses. That process means that you can go broke in the short run, despite having a great long-term strategy.
Therefore instead of just looking at long-term returns you should also look at things like, “What are my returns after 100 trials if I’m unlucky enough to be at the 20th percentile?” There are a number of ways to calculate that. The simplest is to say that if p is your probability of winning, the expected number of times you’ll win is 100p. The variance in a single trial is p(1-p). And therefore the variance of 100 trials is 100p(1-p). Your standard deviation in wins is the square root, or 10sqrt(p(1-p)). From the central limit theorem, at the 20th percentile you’ll therefore win roughly 100p − 8.5sqrt(p(1-p)) times. Divide this by 100 to get the proportion q that you won. Your ideal strategy on this metric will be Kelly with p replaced by that q. This will always be less than Kelly. Then you can apply that to figure out what rate of return you’d be worrying about if you were that unlucky.
Any individual gambler should play around with these numbers. Base it on your bankroll, what you’re comfortable with losing, how frequent and risky your bets are, and so on. It takes work to figure out your risk profile. Most will decide on something less than Kelly.
Of course if your risk profile is dominated by the pleasure of the adrenaline from knowing that you could go broke, then you might think differently. But professional gamblers who think that way generally don’t remain professional gamblers over the long haul.
The reason I brought this up, which may have seemed nitpicky, is that I think this undercuts your argument for sub-Kelly betting. When people say that variance is bad, they mean that because of diminishing marginal returns, lower variance is better when the mean stays the same. Geometric mean is already the expectation of a function that gets diminishing marginal returns, and when it’s geometric mean that stays fixed, lower variance is better if your marginal returns diminish even more than that. Do they? Perhaps, but it’s not obvious. And if your marginal returns diminish but less than for log, then higher variance is better. I don’t think any of median, mode, or looking at which thing more often gets a higher value are the sorts of things that it makes sense to talk about trading off against lowering variance either. You really want mean for that.
The reason why variance matters is that high variance increases your odds of going broke. In reality, gamblers don’t simply get to reinvest all of their money. They have to take money out for expenses. That process means that you can go broke in the short run, despite having a great long-term strategy.
Therefore instead of just looking at long-term returns you should also look at things like, “What are my returns after 100 trials if I’m unlucky enough to be at the 20th percentile?” There are a number of ways to calculate that. The simplest is to say that if p is your probability of winning, the expected number of times you’ll win is 100p. The variance in a single trial is p(1-p). And therefore the variance of 100 trials is 100p(1-p). Your standard deviation in wins is the square root, or 10sqrt(p(1-p)). From the central limit theorem, at the 20th percentile you’ll therefore win roughly 100p − 8.5sqrt(p(1-p)) times. Divide this by 100 to get the proportion q that you won. Your ideal strategy on this metric will be Kelly with p replaced by that q. This will always be less than Kelly. Then you can apply that to figure out what rate of return you’d be worrying about if you were that unlucky.
Any individual gambler should play around with these numbers. Base it on your bankroll, what you’re comfortable with losing, how frequent and risky your bets are, and so on. It takes work to figure out your risk profile. Most will decide on something less than Kelly.
Of course if your risk profile is dominated by the pleasure of the adrenaline from knowing that you could go broke, then you might think differently. But professional gamblers who think that way generally don’t remain professional gamblers over the long haul.