Why it breaks in practice, two hard walls rather than one:
- Table limits. A typical even-money table with a $5 minimum carries a $500 maximum. Step 7 needs $320 (allowed). Step 8 needs $640 (refused). The progression is structurally incapable of completing. You are not risking $5,115 to win $5; you are risking $635 with no recovery mechanism beyond it.
- Bankroll. Absorbing ten consecutive losses requires $5,115 of available funds to secure $5 of profit. The probability of ten consecutive losses on red on a European wheel is (19/37)¹⁰ ≈ 0.13%, roughly once in every 780 attempts. Run the system daily and you will meet it. When you do, one sequence erases hundreds of $5 wins.
Fibonacci. Stakes follow 1, 1, 2, 3, 5, 8, 13, 21, 34 and so on, advancing one step after a loss and retreating two after a win. From a $5 unit: $5, $5, $10, $15, $25, $40, $65, $105, $170. Growth is slower than Martingale, so the table limit arrives later. But a single win no longer clears the full deficit, so recovery needs several wins. The same ceiling applies; it just arrives with a different shape.
D'Alembert. Increase by one unit after a loss, decrease by one after a win. From $5 with $5 increments: loss to $10, to $15, to $20; a win steps back to $15. Far gentler drawdown, far less explosive recovery. The system implicitly assumes wins and losses will roughly balance, which is exactly the gambler's fallacy dressed as arithmetic (Section 6.2).
Flat betting. Identical stake every spin. It is the only approach with an honest, calculable outcome: expected loss = turnover × house edge. $5 × 100 spins = $500 turnover, so −$13.50 expected on a European wheel, or −$26.30 on an American one. It cannot generate a positive expectation, but it also cannot generate a catastrophic single-session loss, and it makes your bankroll requirement predictable in advance. For anyone playing for entertainment rather than profit, it is the defensible choice.
What all four share: they are bankroll-management schemes, not edge-reduction schemes. Choosing a French wheel over an American one improves your expected return by 3.91 percentage points. Choosing Martingale over flat betting improves it by exactly zero.