Reducing Variance in Bets

You have 1000 coins and must place bets repeatedly until you’ve wagered all 1000 coins at least once before cashing out. Each round, you decide how many coins \(x\) to bet. Flipping a fair coin determines the outcome: heads gives you your bet back plus 90% more; tails loses your bet. What’s the optimal betting strategy to maximize your coins by the end?

Answer

Each coin wager has expected profit

\[ 0.5\,(+0.9) + 0.5\,(-1) = -0.05, \]

so on 1000 coins you expect to lose \(1000\times0.05=50\), ending with 950.

Two philosophies for placing your bets:

  • Single-Bet Philosophy: Bet all 1000 coins at once.

    • EV of payoff 950.

    • Variance: very high (either +900 or –1000).

  • Many-Small-Bets Philosophy: Bet 1 coin per round, 1000 rounds.

    • Each bet has (payoff) EV of 0.95 and in total the EV is the same as above, i.e., 950.

    • By the Law of Large Numbers, the sum concentrates near \(950\) and the variance is minimal (the more bets we place the lower the variance).

Conclusion: It is always better to reduce the variance in these situations and hence go with the second strategy.

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