The classic Gambler’s Ruin problem studies a simple setup: you repeatedly place even-money bets and ask how likely it is that you eventually go broke. Real betting strategies don’t look like that. Odds are not even money, and people often bet repeatedly on events where the potential upside is much larger than the downside.
This paper generalises the Gambler’s Ruin problem to that more realistic setting. It derives exact ruin probabilities when bets have asymmetric payoffs and shows how these probabilities depend on both the odds and the bettor’s perceived edge. A key result is that among strategies with an equal edge in terms of expected profit, the one where wins are large relative to losses can have higher probabilities of eventual ruin, sometimes very high. The reason is simple: big wins usually mean low win probabilities, so you can go broke while waiting for the rare win to arrive.
One useful way to think about this is that “positive expected value” is not the same thing as “safe”. With asymmetric bets, you can have a positive edge and still be almost certain to go broke even if you’re just betting the same modest stake if you face a long enough sequence of bets.
Key research finding: With asymmetric odds, ruin probabilities can be high even for strategies with positive expected value, and the classic even-money intuition badly understates the risk of going broke.
Practical advice: If your strategy involves rare big wins, expected returns alone are a poor guide. You need to think explicitly about the probability of ruin, not just the average outcome, and be careful when sizing bets.