Quote from: Luxo42 on March 06, 2018, 09:59:05 AM

Quote from: 01010100b on March 06, 2018, 09:12:22 AM

Quote from: RHavar on March 05, 2018, 01:47:42 AM

However I think he made the same mistake I struggled with too, which was thinking that if the bankroll is -EBG that would allow a player to be +EBG. Fortunately (for investors) this isn't the case, so there's no real abuse avenue.

Can you check where I'm going wrong with this? Here's how I looked at it:

Let BG and PG be two random variables, BG is the house's bankroll growth and PG is the player's bankroll growth (in absolute numbers). Then:

BG + PG = 0 (money only moves back and forth between the house and the player, so a loss for one is a gain for the other and vice versa)
E[BG + PG] = E[0] (taking the expected value of both sides)
E[BG + PG] = 0 (expected value of a constant is that constant)
E[BG] + E[PG] = 0 (by linearity of expected value)
E[BG] = -E[PG]

Which leads to the player's EBG being the opposite of the house's EBG, so if the house is at -EBG then the player is at +EBG and vice versa. Perhaps by EBG is meant the median bankroll growth and not the mean bankroll growth? In which case the linearity condition above wouldn't apply.

No. That is for expected value only.

Yes, the expected value of the bankroll growth in absolute numbers. BG and PG are random variables.