D’Alembert: adding one after a loss

D’Alembert increases an even-money stake by one unit after a loss and decreases it by one after a win, with a one-unit minimum. Stake growth is linear, but a long losing run still creates a rising bill.

Follow the stake as well as the result

Start with one unit and use the outcomes loss, loss, win, win. The stakes are 1, 2, 3, 2. The cycle earns −1−2+3+2=+2 units. A different order can produce a different result: win, win, loss, loss uses 1, 1, 1, 2 and finishes −1 unit. Both paths contain two wins and two losses. The floor makes the order matter.

The audited native named plan follows this one-unit floor. A real demonstration should also stop before a required stake exceeds the wallet or table maximum.

Losing runs are cheaper than doubling, but still grow

Ten consecutive losses at 1 through 10 units consume 55 units. The next stake is 11. Ordinary Martingale loses 1023 units across its first ten failed bets. That comparison describes stake exposure; it does not establish a profitable outcome for D’Alembert. A progression that wagers less also has less money at stake on a later winning spin.

Choose a fair comparison

Compare both systems on the same wheel, outcomes and unit value. Report total wagered beside ending balance. A run that stops at a limit should stay in the results rather than disappear as an unfinished cycle. The strategy simulator gives a bounded comparison; flat betting provides a constant-stake reference.

D’Alembert is often described through balanced win/loss examples. The two paths above show why that description needs its stake rules and order. Read roulette variance for the distinction between changing the shape of results and changing expected return.