Editorial team: Why does the D’Alembert system still lose in the long run?

The D’Alembert system does not change the expected value of the underlying wager. It changes how much is staked after wins and losses.

The conditional EV argument needs clear assumptions: each wager has the same fixed conditional expected profit −h per unit staked, where h > 0 is the house edge, regardless of prior history; the nonnegative stake is chosen before the next result; and the session has finite expected turnover.

Under those assumptions, each unit staked has expected profit:

−h

For a stake s, the conditional expectation for that next wager is:

expected profit = −h × s

Summed over the finite expected turnover, expected total profit is:

−h × expected total amount wagered

A regulator explanation of return-to-player and house-edge style averages distinguishes long-run expected return from what happens in a particular session: https://www.gamblingcommission.gov.uk/public-and-players/guide/return-to-player-how-much-gaming-machines-payout

Simple hypothetical example

Suppose an independent even-money wager wins 49% and loses 51%. The expected profit per 1 unit staked is:

0.49 × 1 − 0.51 × 1 = −0.02 units

That is a 2% loss per unit staked. If a D’Alembert sequence produces 1,000 units of expected turnover, the expected profit is:

−0.02 × 1,000 = −20 units

The progression can create many small recoveries and occasional larger drawdowns, but it does not turn that assumed negative unit expectation into a positive one.

For a fuller strategy overview, see the ToolsGambling D’Alembert article: https://toolsgambling.com/blog/dalembert-strategy

Limitations: this does not mean every session loses, and the assumption is not a statement about every game or rule set. Some sessions win. Bankroll limits, table limits and finite time affect outcomes, but they do not create a magic recovery guarantee in a negative-expectation wager.