Does the D’Alembert system work?

How D’Alembert is played

1. Start at one unit ($5) on red.
2. After a loss, raise the stake by one unit.
3. After a win, lower it by one unit (never below one).

What 20,000 sessions show

Sessions that finished ahead were up $207.85 on average; sessions that finished behind were down $943.16. 28.4% of sessions ended because the next stake no longer fit the bankroll or the table limit.

D’Alembert assumes wins and losses will even out, so each win is meant to recover the matching loss with one unit to spare. On a wheel where losses outnumber wins 19 to 18, the stake drifts upward over a long session and the bets that lose are, on average, larger than the bets that win.

The median session lasted 94 spins. The median longest losing run was 6 spins in a row (the worst of 20,000 was 19), and one session in ten put at least $110 on the table in a single spin.

European versus American wheel

The same 20,000 seeds, replayed on a double-zero wheel. The second zero roughly doubles the edge to $5.26 per $100 wagered. Measured return goes from −$2.87 to −$5.24 per $100, and the mean session from −$119 to −$240.

The code that produced these numbers

This Lua is the strategy as Engineer mode runs it, inside the same sandbox and kernel. A test in the repository replays it against the bulk generator on fixed seeds and fails the build if the two ever disagree by a single dollar.

-- D’Alembert: add one unit to the stake after a loss, remove one after a win.
-- Session rules shared by every 0xroulette.com/systems page:
-- $5 unit, $500 table max, walk away at +$200, and stop the moment
-- the next stake can't be covered by the bankroll or the table limit.
local UNIT, TABLE_MAX, TARGET = 5, 500, 200

local level

local function reset() level = 1 end

local function place(ctx)
  return { { type = "red", amount = UNIT * level } }
end

local function settle(result)
  if result.net > 0 then level = math.max(1, level - 1) else level = level + 1 end
end

-- Shared wrapper: enforces the session rules around place/settle.
strategy = {}
local start, stopped

function strategy.setup(ctx)
  start, stopped = ctx.bankroll, false
  reset()
end

function strategy.bet(ctx)
  if stopped or ctx.bankroll >= start + TARGET then
    stopped = true
    return {}
  end
  local bets, stake = place(ctx), 0
  for _, b in ipairs(bets) do
    if b.amount > TABLE_MAX then stopped = true end
    stake = stake + b.amount
  end
  if stake > ctx.bankroll then stopped = true end
  if stopped then return {} end
  return bets
end

function strategy.on_spin(ctx, result)
  if not stopped then settle(result) end
end

Session rules

Limitations

One bankroll, one unit size, one table limit, and one walk-away target. A bigger bankroll or a higher limit moves where a progression breaks; it does not change the return per dollar wagered. Spins are fair and independent, as on a correctly functioning wheel or a certified RNG. Physical wheel bias is a separate question, covered on Physics.

Related

• Does the Contra D’Alembert system work?
• Does the Fibonacci roulette system work?
• Does Oscar’s Grind work?
• Study: Five strategies on the same seeds
• All 20 systems, compared on the same seeds