2. Maximize Your Vegas Winnings! You are in Vegas with $10,000. This is your entire bankroll. The game is simple. I flip a fair coin. You may call heads or tails. If you call correctly, you gain 50% of the amount you bet. If you call incorrectly, you lose 40% of the amount you bet. My game has two unusual conditions: 1. If you choose to play, you must play 100 rounds of the game. 2. You must decide now what percent of your bankroll you play each time. You may choose to bet anywhere from 0 to 100 percent of your current/remaining bankroll. The percent of your remaining bankroll that is bet remains fixed for all 100 rounds. Assume you are a perfectly rational economist who wants to maximize compound growth. What percent of your bankroll should you bet? Submitted by tartle · Added 4 July 2026 · Updated 11 July 2026 Show Hint Show Solution Show Game Open Game Hint: Think like Kelly! Solution: For this you need to use the Kelly formula. Let: f = fraction of bankroll bet Win probability: p = 0.5 Lose probability: q = 0.5 Win return on bet: b = 0.5 Loss rate on bet: a = 0.4 The bankroll multiplier is: Win: 1 + bf = 1 + 0.5f Loss: 1 - af = 1 - 0.4f To maximize compound growth, maximize expected log growth: G(f) = p ln(1 + bf) + q ln(1 - af) Plug in the numbers: G(f) = 0.5 ln(1 + 0.5f) + 0.5 ln(1 - 0.4f) Take derivative and set to zero: G'(f) = 0.5 × 0.5 / (1 + 0.5f) - 0.5 × 0.4 / (1 - 0.4f) G'(f) = 0.25 / (1 + 0.5f) - 0.20 / (1 - 0.4f) Set equal to zero: 0.25 / (1 + 0.5f) = 0.20 / (1 - 0.4f) Cross-multiply: 0.25(1 - 0.4f) = 0.20(1 + 0.5f) 0.25 - 0.10f = 0.20 + 0.10f 0.05 = 0.20f f = 0.25 So: Optimal bet = 25% of current bankroll. Maximize Your Vegas Winnings One bet size, locked in for 100 rounds. Same $10,000, same fair coin. Now you win +50% of your stake on a right call and lose −40% of your stake on a wrong one. You must play all 100 rounds and pick one fixed fraction of your remaining bankroll to wager every time. A rational player maximizing compound growth bets what? Fraction of bankroll you stake each round 60% Send 10,000 gamblers Reveal the optimum Expected (average) wealth Typical (median) wealth Your bet Starting $10,000 Expected wealth predicts — — Typical wealth predicts — — Finished ahead of $10k — of 10,000 gamblers Effectively wiped out — left with under $100
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