Thought experiment: Suppose you bet 100% of your bankroll every round. If at any point you lose a round, your bankroll is now $0. Any money you made from former

Thought experiment: Suppose you bet 100% of your bankroll every round. If at any point you lose a round, your bankroll is now $0. Any money you made from former rounds is for naught. Whoops. bankroll_final = (bankroll_initial)(round_1)(round_2)(round_3)(...)(round_n) $0 = (bankroll_initial)(210% )(210% )(210% )(...)(0% ) Taking the "non-log" Expected Value would be optimal if your bankroll were "renewed" to the same constant each round. Because the outcome of each individual round would be independent from other rounds. Bankroll_final = (bankroll)(round_1) + (bankroll)(round_2) + (...) + (bankroll)(round_n) But since the outcome of each round depends on previous rounds, we want to optimize for Expected Value of Growth. For which we'll use a geometric mean rather than an arithmetic mean. Also, EV[bankroll_final] = EV[(bankroll_initial)(round_1)(round_2)(round_3)(...)(round_n)] is equivalent to EV[bankroll_final] = EV[(bankroll_initial) e^ln(r1 + r2 + r3 + (...) + r_n)] which allows us to discuss wagers in terms of e^(x) and in terms of growthrates.