This problem was posted in Scientific American (vol. 321.5, Nov 2019, p. 73), and it was troubling.
The game:
We flip a fair coin. If we flip heads we gain 20% of our bet If we flip tails we lose 17% of our bet. Starting bankroll: $100
Stipulations: We must bet all the chips we have, cannot reload, and we must play at least 10 flips.
Note- There's no minimum bet, in the sense that we can keep playing this game with an infinitesimally small chip stack. Since we're always betting a fraction of our bankroll, there's no risk of ruin.
The expected value is net positive. Using the first flip as an example: EV = (0.5 * $20) - (0.5 * $17) = + $1.50
For 10 flips considering all outcomes: EV = +16.05% https://docs.google.com/spreadsheets/d/15nXStFnsEHFU938erWaCKDVMmFdaOVeHNJ-taQZLcJs/edit?usp=drivesdk
However, there's another way to look at this. When we win our bet is multiplied by 1.2. When we lose our bet is multiplied by 0.83.
Let's say we win one, lose one: 1.2*0.83 = 0.996
If we win 5 flips and lose 5 flips in any order: 1.2 x 1.2 x 1.2 x 1.2 x 1.2 x 0.83 x 0.83 x 0.83 x 0.83 x 0.83 = 0.9802, for a net loss of -$1.98.
Losses "hit harder" than wins. From this perspective it looks like a disadvantageous game.
Put technically, the geometric mean is less than 1, which means we expect our bankroll to shrink on average.
In this game the Kelly Criterion says we should risk no more than 7.5%, however we're forced to risk more than double that, 17%. It's a well known principle that betting more then twice Kelly will result in a shrinking bankroll. This is just another way to say the Geometric mean of growth is less than 1.
Scientific American claims the casino will make money over time. They claim that this holds true even if you flip 100 or 10,000 times or more.
I can't wrap my head around the fact that a +EV bet could lose over time in a game with no risk of ruin. I suspect this has to do with "volatility drag". While I can understand the math, I can't intuitively wrap my head around it.
So my question is, should you play this game?
from Hot Weekly Questions - Mathematics Stack Exchange
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