The Wealth Paradox of the Coin Toss Game: Why High Expectations Fail to Combat the Fate of Zero?
Tossing a coin doubles the asset on heads, while it shrinks by 60% on tails. Looking at a single game, the expected value is as high as 20%, making it a 'theoretical money printer.' However, simulations show that after 25,000 people each toss the coin 1,000 times, almost everyone ends up at zero. This seemingly contradictory result hides a deep game between the arithmetic mean and the geometric mean.
The 'Trap' of Expected Value
The 20% positive expected value of a single coin toss is calculated based on the arithmetic mean: (100% - 60%) ÷ 2 = 20%. But in reality, asset changes follow a multiplicative effect—each gain or loss is based on the current asset. In the long run, the geometric mean (which measures compounding effects) is key, and the geometric mean of this game is negative, meaning compounding will gradually erode the assets.
The 'Illusion' of Extreme Probabilities
To break even after 1,000 tosses, one needs at least 570 heads and 430 tails, which is itself a low-probability event. More critically, the high expectation of the arithmetic mean heavily relies on extreme rare scenarios like 'consecutive heads.' Simulations show that after 1,000 tosses, all positive expected values are concentrated in 0.0001% of extreme lucky individuals, which for the vast majority is merely an illusion.
The 'Invisible Killer' that is Overlooked
This phenomenon is known as the 'Jackpot Paradox': when high expectations are tied to extremely small probabilities, high-probability events point towards losses instead. Physicists refer to this as the 'Ergodicity Problem'—the average result of a group cannot be equated to the long-term results of individuals; traders call it 'Volatility Drain'—excessive fluctuations can turn theoretical positive expectations into actual paths to zero.
What this coin toss game reveals is the non-linear relationship between risk and return: when chasing high expectations, if one ignores the high-probability losses under extreme volatility, it may ultimately lead one towards zero in the illusion of 'theoretical profit.'