Here’s an example of a game to start us off:
Imagine in this game you have a 70% chance of winning 200% (3x) and 30% chance of losing 99%. The game’s arithmetic expected value (EV) is quite high, at 2.103.
Now imagine playing this game 10 times in a row and fully reinvesting your returns each time. Guess what are your chances of losing 95% of your money by the end of the 10 rounds? A whopping 61% of the time!
How could such a high arithmetic EV lead to such a terrible probabilistic outcome?
The answer lies in the difference between the arithmetic EV and the geometric EV. In a game where you’re playing rounds sequentially and fully reinvesting returns each round, even though each individual round’s EV is the arithmetic EV, your overall EV is the geometric EV.
What’s the geometric EV?
The equation is outcome_1 ^ chance_1 * outcome_2 ^ chance_2.
For the game above its geometric EV is 3^70% * 0.01^30% = 0.54.
Now you can see, 0.54 is a far cry from 2.103 and this is why after 10 games, you have such a high chance of losing basically your entire bank roll.
When you are growing your wealth, to enjoy the vast benefits of exponential growth, you have to reinvest more and more of your wealth over time.
So how does one do so without subjecting yourself to the geometric EV and instead benefit from the arithmetic EV?
The answer to this question lies in fractionalizing your bets, aka “don’t put all your eggs in one basket”.
The more you fractionalize your bets, and the more uncorrelated they are, the more your overall EV approaches the average arithmetic EV across all your bets.
