Fibonacci, Labouchere, D'Alembert

The Shared Flaw Behind Three Different Systems

Start with the arithmetic that makes all three systems equivalent in outcome. Total expected loss on any session of roulette equals total amount wagered multiplied by the house edge. On a European single-zero wheel that's 2.70% of everything you stake, not of your starting bankroll but of every chip that crosses the felt in either direction. A system that changes your bet sizes between spins changes when and how much you wager, not the mathematical relationship between what you put down and what you get back. The casino collects 2.70% of each chip regardless of which spin it appears on or how it got there.

This makes all progression systems mathematically equivalent to flat betting at the same average stake. The only thing a system can change is variance: how smooth or rough the path to that expected loss is, and what maximum bet size you'll need at the worst moment. None of the three systems discussed here reduces expected loss. Two of them substantially increase it by increasing total wagered action. That's the baseline from which to evaluate everything below.

Fibonacci: Slow Growth, Same Drift

The Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55..., applies to roulette as follows: move one step forward in the sequence after a loss, two steps backward after a win. The appeal is that after a win you recover two losses for each win, and the bet sizes grow more slowly than a Martingale. After six consecutive losses, Fibonacci has you betting 13 units where the Martingale would demand 64. The bust rate at the table limit is lower.

The problem is that the wins are proportionally smaller, the recovery mechanism requires winning streaks to meaningfully outnumber losing streaks, and if you reach a long losing run before a recovery sequence, you've wagered substantially more in total than flat betting would have cost. Consider a ten-loss streak followed by ten wins from the starting position. Flat betting at 1 unit: wagered 20 units, expected loss 0.54 units. Fibonacci across those 20 spins: total wagered 154 units, expected loss 4.16 units. Same number of bets, same sequence of outcomes, nearly eight times the expected loss because the system inflated the wagers. The Fibonacci is slower than the Martingale but reaches the same place for the same reason.

Labouchere: The Cancellation System and Charles Wells

The Labouchere, or cancellation system, works differently. You write a sequence of numbers, say 1-2-3. Your bet is the sum of the first and last numbers: 1+3=4. A win: cross off both numbers. A loss: add the losing bet to the end of the sequence. Continue until all numbers are crossed off, at which point your profit equals the sum of the original sequence (6 units in this case). The sequence itself can be any length and any numbers; the profit target is the sum of the starting line.

The system's appeal is that it requires more wins than losses to complete (a win cancels two numbers; a loss adds one), but a single prolonged losing run can extend the sequence to extreme lengths. After several losses, you may need a bet fifty times your original unit to continue the sequence. Table limits terminate the sequence prematurely and lock in a large loss.

The Labouchere is often associated with Charles Wells, the "man who broke the bank at Monte Carlo" in 1891, celebrated in the music hall song of the same era. The Wikipedia account of Wells notes that contemporary investigators and later historians concluded his wins were attributable to extraordinary statistical variance rather than any system. He won approximately one million francs over several sessions, then returned and lost the majority of it back. Wells was subsequently imprisoned for unrelated fraud, which coloured the popular interpretation of his gambling episodes as part of a larger con. The system associated with his name doesn't explain his wins; variance does.

The Casino de Monte-Carlo reportedly adjusted its betting limits and procedural controls partly in response to Wells's run, per contemporary newspaper accounts. The casino's French roulette grande table still operates today with a 5-minute minimum and 2,000-euro maximum per bet, per the Casino de Monte-Carlo's own information pages.

D'Alembert: The False Equilibrium

The D'Alembert system, named after the 18th-century French mathematician Jean le Rond d'Alembert, increases bets by one unit after a loss and decreases by one unit after a win. The theoretical basis is the Equilibrium Principle: over time, wins and losses will tend to balance, so after a loss you're "due" a win and after a win you're "due" a loss. Both premises are false as applied to roulette.

D'Alembert himself was a distinguished mathematician, but his 1754 work in which he argued that after a sequence of tails the probability of heads increases was wrong, as later mathematicians demonstrated. Each roulette spin is an independent event. The wheel has no memory of previous outcomes. The probability of red on the next spin is 18/37 regardless of how many blacks have preceded it. This is the Gambler's Fallacy, extensively documented in the Conversation's December 2016 explainer.

The D'Alembert does produce smaller swings than the Martingale and smaller losses in a losing run than the Fibonacci, because bet increases are linear (add 1) rather than multiplicative (double) or Fibonacci sequential. But it also recovers more slowly. A run of ten losses followed by ten wins returns you to your starting position only if the ten wins come at larger bet sizes than the ten losses, which requires the wins to happen after the losses, which you can't guarantee. If you win ten first and then lose ten, you're behind by more than flat betting would have cost.

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