Coin Toss Game and Mathematical Principles

Overview

• Game starts with $100.
• Coin toss mechanics:
  • Heads: increases wealth by 80% (multiplies by 1.8).
  • Tails: decreases wealth by 50% (multiplies by 0.5).

Game Simulation

• Simulate 1 million players over 50 rounds.
• Graph shows average wealth grows exponentially.
• Median and mode drop to $7.2.

Non-Ergodic System

• Definition: Population average is different from the median outcome.
• Average value misleads about individual outcomes.

Visualizing Outcomes

• Wealth visualization after each flip.
• Example: flipping 1 heads and 1 tails leads to a drop to $90.
• Average outcome pulled up by lucky outliers.

Logarithmic View

• Sequences of flips lead to different wealth outcomes.
• Most common outcome: half heads and half tails.
• Median and mode reflect the center and most common outcomes.

Just One More Paradox

• Despite favorable odds, the overall outcome is negative.
• Multiplicative nature of game leads to less upside at lower wealth levels.

Strategy Adjustments

• Changing to an additive betting strategy by betting a fixed amount ($50).
• New mode slopes upwards, equal to average wealth.
• Questions raised:
  • Bet size adjustments when below $50 or above $1000.

Optimal Betting Strategy

• Consider betting a fraction of wealth (1/5 or 1/10).
• Mode growth rate derived from betting fractions.
• Geometric mean introduced for evaluating growth rate.

Maximizing Growth Rate

• General formula introduced:
  • F: fraction of wealth
  • B: gain per heads
  • A: loss per tails
  • P/Q: probabilities of heads/tails
• Graph of betting fraction vs. mode growth rate.
  • Betting 0: no change, growth rate = 1.
  • Betting 1: total loss, growth rate < 1.
• Highlighted segment shows potential for profit between 0 and 1.

Finding Optimal Fraction

• Calculus used to find peak growth rate, resulting fraction = 0.375.
• Mode growth rate at this fraction = 1.028 per flip.

Final Simulation

• Simulation with million players betting 0.375 of wealth.
• Results show median increases steadily, thanks to the Kelly Criterion.

Conclusion

• Kelly Criterion: recommends optimal betting fraction to maximize growth.
• Key concepts learned: probability, paradoxes, and maximizing strategies.
• Acknowledgments for resources used in the video.