Not in general. As you say, GM requires positive numbers, but there’s a reason for this: imagine GM as log-scaling everything and then performing AM on the results.
So to get the GM of 10 and 1000:
Realize that 10 = 10^1, 1000 = 10^3
Then average 1 and 3 to get 2.
So your result is 10^2=100.
But now notice that:
0.01 is 10^-2
0.00001 is 10^-5
0.0000000001 is 10^-10
0 is 10^[negative infinity]?
-1 is...uh...
and so the GM of 1 million and 0.00000000000000000000000001 is 0.00000000000001, and the GM of 1 billion and 0 is 0. This won’t really lend itself to calculating a GM of a list including a negative number.
One thing you can do, though, which makes sense if you are e.g. calculating your utility as log(your net worth) in various situations, is calculate the GM of [your current net worth + this value].
For instance, if you are considering a gamble that has a 50% chance of gaining you $2000 and a 50% chance of losing you $1000:
If your net worth is $1000, this replaces $1000 with a 50% chance of $3000 and a 50% change of $0. Since GM(3000, 0) = 0, this is worse than just staying with the $1000 .
If your net worth is $2000, this replaces $2000 with a 50% chance of $4000 and a 50% chance of $1000. Since GM(4000, 1000) = 2000, you are indifferent.
If your net worth is $4000, this replaces $4000 with a 50% chance of $6000 and a 50% chance of $3000. Since GM(6000, 3000) ~= 4242, this is better than staying with the $4000.
The geometric expected value for the +100% or −60% gamble in the post is straightforward.
If you gain 100% and then lose 60% (or lose 60% and then gain 100%), overall you’ll end up down 20% from where you started since (1 + 100%) * (1 − 60%) = 2 * 0.4 = 80% = 1 − 20%. This loss is what’s expected over two iterations, so the loss over one iteration is sqrt(80%) = 89.44% = 1 − 10.56%, so the expected loss is that 10.56% of your starting capital per flip.
Not in general. As you say, GM requires positive numbers, but there’s a reason for this: imagine GM as log-scaling everything and then performing AM on the results.
So to get the GM of 10 and 1000:
Realize that 10 = 10^1, 1000 = 10^3
Then average 1 and 3 to get 2.
So your result is 10^2=100.
But now notice that:
0.01 is 10^-2
0.00001 is 10^-5
0.0000000001 is 10^-10
0 is 10^[negative infinity]?
-1 is...uh...
and so the GM of 1 million and 0.00000000000000000000000001 is 0.00000000000001, and the GM of 1 billion and 0 is 0. This won’t really lend itself to calculating a GM of a list including a negative number.
One thing you can do, though, which makes sense if you are e.g. calculating your utility as log(your net worth) in various situations, is calculate the GM of [your current net worth + this value].
For instance, if you are considering a gamble that has a 50% chance of gaining you $2000 and a 50% chance of losing you $1000:
If your net worth is $1000, this replaces $1000 with a 50% chance of $3000 and a 50% change of $0. Since GM(3000, 0) = 0, this is worse than just staying with the $1000 .
If your net worth is $2000, this replaces $2000 with a 50% chance of $4000 and a 50% chance of $1000. Since GM(4000, 1000) = 2000, you are indifferent.
If your net worth is $4000, this replaces $4000 with a 50% chance of $6000 and a 50% chance of $3000. Since GM(6000, 3000) ~= 4242, this is better than staying with the $4000.
The geometric expected value for the +100% or −60% gamble in the post is straightforward.
If you gain 100% and then lose 60% (or lose 60% and then gain 100%), overall you’ll end up down 20% from where you started since (1 + 100%) * (1 − 60%) = 2 * 0.4 = 80% = 1 − 20%. This loss is what’s expected over two iterations, so the loss over one iteration is sqrt(80%) = 89.44% = 1 − 10.56%, so the expected loss is that 10.56% of your starting capital per flip.