The Dilemma of the Button: A Decision-Making Analysis for Gamblers and Investors

Situation: Infinite times you can press a button that has a 99% chance of doubling your money and a 1% chance of wiping it out. You start with $1. How many times should you press the button?

Expected Value Calculation

Step 1: Pressing the Button Once

If you win (99% probability), the outcome is $2. If you lose (1% probability), the outcome is $0.

The expected value (EV) for pressing the button once can be calculated as:

EV 0.99 * 2 0.01 * 0 $1.98

Since the expected outcome ($1.98) is greater than your starting value ($1), pressing the button once appears to be a favorable decision.

Pressing the Button Multiple Times

Step 2: Compounded Outcomes

The potential outcomes compound as you press the button multiple times. Each press multiplies your current amount by 2 with a 99% chance and resets everything with a 1% chance.

Step 3: Risk of Losing Everything

The risk of losing everything increases with each press. For n presses, the probability of losing everything at least once is given by:

1 - 0.99n

This formula helps you understand how the risk escalates with each additional press.

Decision-Making on the Number of Presses

Risk Tolerance

For Risk-Averse Individuals:

If you are highly risk-averse, you might choose to press the button fewer times to minimize the risk of total loss. Lowering the number of presses ensures a safer outcome but limits potential returns.

For Risk-Taking Individuals:

If you are willing to accept higher risk for potentially higher returns, you might press the button more times. More presses increase the chance of substantial gains, but also the chance of total loss.

Expected Growth

The more times you press the button, the greater the expected growth from a multiplicative perspective. However, each pressing also amplifies the risk.

Strategic Approach

In practice, many would suggest starting with a small number of presses, such as 1-3, to test the waters and reassess based on outcomes and your risk tolerance.

Each additional press increases the risk of a total loss. Therefore, carefully consider your risk tolerance and make informed decisions.

Conclusion

If you press the button once, the expected value suggests it is a good decision, given the higher than initial expected return.

However, if you press the button multiple times, you need to balance the potential for significantly higher returns against the increasing risk of total loss.

Adopt a strategic approach, starting cautiously and adjusting your strategy based on your outcomes and comfort level with risk.