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By graphing the relationship between the betting fraction and mode growth rate, it becomes evident that the highest growth occurs at a specific point. Using calculus to derive the peak value reveals that betting 37.5% of one’s wealth yields the most significant growth rate—1.028 per coin flip.
Conclusion: The Intersection of Probability and Decision Making
After re-simulating the scenario with a million players betting only 37.5% of their wealth, the median outcomes improve, showcasing the effectiveness of the Kelly Criterion. This intricate interplay between probability, wealth, and decision-making underscores the challenges present within non-ergodic systems.