D'Alembert, Labouchere, and Paroli: Three More Betting Systems Explained
D'Alembert, Labouchere, and Paroli are three more even-money betting systems that reshape the rhythm of a session without touching the numbers underneath.
D'Alembert, Labouchere, and Paroli are three more even-money betting systems that reshape the rhythm of a session without touching the numbers underneath. The first two are negative progressions: raise the bet after a loss, hoping the next win recovers ground. Paroli runs the opposite way, a positive progression that raises the bet after a win and lets a streak carry itself. D'Alembert nudges the stake up or down by a single unit, a gentler slope than doubling. Labouchere, the cancellation system, works from a written list of numbers, betting the sum of the two ends and crossing them off as the list clears. Paroli, sometimes called the reverse Martingale, doubles after a win, usually chasing three in a row, then resets to one unit. Each promises a different shape to the evening, steadier for one, streak-hungry for another. None of them touch the house edge or break the independence of each spin, so the arithmetic under the felt never changes. What changes is only the pattern of how wins and losses land, worth understanding before any of the three ever gets used at a table.
What is the real difference between a negative and a positive progression?
A negative progression raises the bet after a loss, chasing recovery on the next win. A positive progression raises the bet after a win, letting a streak fund itself. D'Alembert and Labouchere are negative; Paroli is positive. Both directions leave the underlying odds exactly where they started.
Negative progressions raise the bet after a loss, chasing a win that feels overdue. Positive progressions raise the bet after a win, riding a streak already in motion. D'Alembert and Labouchere lean negative; Paroli leans positive, though neither lean changes what the wheel or the deck produces next.
How does the D'Alembert system work?
Named for the French mathematician, D'Alembert raises the bet by one unit after a loss and lowers it by one unit after a win. It is a linear climb, not a doubling one, built on the idea that wins and losses roughly balance out over a session.
Start at one unit. Lose, and the next bet rises by one unit; win, and it drops by one unit, a gentle staircase rather than a doubling one. The assumption is that wins and losses roughly balance across a session, leaving a small profit once the ladder settles. A genuine losing run still climbs regardless, one unit at a time, and table limits or a finite bankroll can outrun it. The house edge sits underneath the staircase, untouched by how the bet size moves.
How does the Labouchere, or cancellation, system work?
Labouchere starts with a written sequence of numbers. Each bet equals the sum of the first and last numbers on the list; a win crosses both off, a loss adds the lost amount to the end. Clearing the whole list marks a session profit.
Write down a short sequence, say 1, 2, 3, 4, before the first bet. The stake is always the sum of the two end numbers. Win, and both ends get crossed off, shortening the list; lose, and the amount lost is added to the end, lengthening it. So long as the list eventually empties, the session nets a profit equal to the original sum, but a losing run lengthens the list and pushes the next stake higher, escalating toward table limits or the edge of a bankroll.
A loss lengthens the list; only clearing it pays off.
How does the Paroli, or reverse Martingale, system work?
Paroli flips the Martingale: instead of doubling after a loss, it doubles the bet after a win, usually aiming for three consecutive wins before resetting to one unit. Losses stay capped at a single unit each time.
Start at one unit. Win, and the next bet doubles, typically toward a target of three wins in a row; hit the target, or lose along the way, and the sequence resets to one unit. Because a loss simply ends the sequence at its current size, no single loss grows the way it can under a negative progression. Paroli still depends on streaks appearing, and each bet remains just as independent of the last as ever; gains compound while a streak lasts, and the downside from any break stays limited to one unit.
Do D'Alembert, Labouchere, or Paroli change the house edge?
No. None of the three alters the house edge or the independence of individual outcomes. Whatever a system does to bet sizing, the underlying probability of each spin, hand, or roll stays exactly the same as flat betting.
A roulette wheel does not remember the last ten spins; a reshuffled shoe carries no memory of the previous hand. Changing the size of the next bet changes nothing about what that next outcome is likely to be. On European roulette, an even-money bet like red or black carries a house edge of about 2.7 ENT per 100 staked, and that figure holds whether the stake sits at one unit or ten, whatever staircase or list brought it there. Expected value stays negative across all three systems, at exactly the rate flat betting would produce.
The wheel keeps no memory of the last ten spins.
Negative progression or positive: what is actually being risked?
Negative progressions risk larger losses while chasing recovery, since the bet keeps climbing after every loss. Positive progressions risk giving back money already won, but they cap the downside at a single unit per broken streak.
Laid side by side, the two directions trade one risk for another rather than removing risk altogether.
- Negative progressions (D'Alembert, Labouchere): the bet grows after a loss, compounding stakes just as the bankroll comes under pressure.
- Positive progressions (Paroli): the bet grows after a win, so any single loss costs only the one unit already in play.
- D'Alembert climbs in a straight line; Labouchere can climb faster, since a loss adds a whole new number rather than one fixed step.
- Paroli caps the downside, but its upside depends on a streak arriving, no more likely than under flat betting.
- Neither direction changes what the next outcome is likely to be.
If the odds never move, what do these systems actually change?
They change the shape of the ride, not the result. Bet sizing rearranges how wins and losses feel across a session; a gentle climb, a clearing list, or a compounding streak each produce a different emotional texture from flat betting, even when the long-run math is identical.
A betting system does not find an edge that was not there. It rearranges the timing and size of wins and losses so a session feels steadier, or like a comeback in progress, or like a hot streak being ridden, depending on which system is used. That reshaping is real: a linear climb, a clearing list, or a compounding streak each offer a genuine difference in experience. What none of them offers is a better long-run outcome, since the independence of each bet and the house edge built into the game sit outside anything a system can touch.
So how should a player choose between the three?
Choose based on the temperament wanted at the table, not on an expectation of beating the game. A steadier climb, a list to clear, or a streak to ride each shape the session differently; over time, each returns the same negative expectation as flat betting.
D'Alembert suits a player who wants a gentle, linear rhythm and can tolerate a slow climb during a rough patch. Labouchere suits a player who likes the structure of a written plan and the satisfaction of watching it clear, provided the length of a losing run is respected before it nears table limits. Paroli suits a player who would rather let winning streaks do the work, accepting that a broken streak only ever costs one unit. None of the three beats the game; each simply decides what the evening feels like while playing it.
The house always knows this
D'Alembert, Labouchere, and Paroli reshape a session's feel, never the house edge or the independence of the next bet.
Frequently asked
Is D'Alembert safer than the Martingale?
It is gentler, since the climb after a loss is linear rather than doubling, so the bet size grows more slowly during a rough stretch. It is not safer in the sense of changing outcomes: the house edge and the independence of each spin remain exactly the same either way.
Can the Labouchere list ever fail to clear?
Yes. A sustained losing run keeps adding numbers to the end of the list, pushing the required stake higher with each loss. Table limits or a limited bankroll can stop a player from continuing long before the list would have cleared on its own.
Why does Paroli usually target three wins in a row?
Three wins is a common convention, not a rule the game enforces. It sets a point to bank the compounding gains and reset to one unit, capping how much of a streak's growth gets risked back before the sequence starts again from the beginning.
Do these systems work better on even-money bets specifically?
They are built around even-money bets like red and black or high and low because the unit-based math is simplest there. The same logic could apply to other wagers, but the systems described here assume a roughly balanced payout structure at each step.
Does switching between these systems mid-session change anything?
No. Swapping from D'Alembert to Paroli, or resetting a Labouchere list midway through a session, only changes the pattern of stakes going forward. Each new bet still remains independent of the last, and the house edge underneath every wager stays exactly where it started out.
Sources & further reading
ENTBlog is educational. Every casino game carries a house edge, so the mathematically expected result of play is a net loss over time. Play for entertainment, within limits you set in advance. Nothing here is financial advice or a promise of winnings.