Martingale Betting Strategy

Person A and Person B play a coin-tossing game. Person A bets $$x$$ dollars on either heads or tails. Person B then flips a fair coin. If the outcome matches A's guess, A receives $$2x$$ dollars; if not, A loses his bet. Person A adopts the following strategy: he starts by betting 1 dollar ( $$x := 1$$ ). If he wins, he continues to bet 1 dollar; if he loses, he triples his previous bet ( $$x := 3x$$ ). It appears that if A wins at any point, he makes a profit, suggesting that he will always eventually profit. Is there any flaw or problem with this strategy?

Answer

The flaw in this strategy is that it relies on the unrealistic assumption of an infinite bankroll and the absence of betting limits. In a real-world setting:

Person A has a finite amount of money, so a long losing streak could force him to exhaust his funds before recouping his losses.
Casinos and betting establishments impose maximum bet limits, which prevent continuous tripling of the bet.
Even though a win yields a profit, the probability of a long losing streak is non-negligible (higher compared to the common intuition), and the exponential growth of the bet size in such a streak can result in catastrophic losses.

Notes and comments

Comment 1: If we would instead only double our bet each round, then when we eventually win on the $$k$$ th round, our total winnings will be $$2^k$$ dollars while our cumulative wager is $$2^k - 1$$ dollars. Thus, our net profit is

\[ 2^k - (2^k - 1) = 1, \]

assuming we start with a $1 bet. More generally, if the initial stake is $$x$$ dollars, then upon winning, the net profit will be $$x$$ dollars.