Does the Paroli system work?

How Paroli is played

1. Bet one unit ($5) on red.
2. After a win, double the stake.
3. After three wins in a row, or any loss, go back to one unit.

What 20,000 sessions show

Sessions that finished ahead were up $154.74 on average; sessions that finished behind were down $227.78. Almost no session ran out of money or hit the table limit; the losses came from steady erosion instead. 79.0% reached the 500-spin cap without hitting the target or busting.

Paroli caps both ends: the most it can risk on one spin is $20, and a completed three-win streak pays seven units. Hitting three in a row on an 18/37 bet happens about 11.5% of the time, so most cycles end in a one-unit loss.

The median session lasted 500 spins. The median longest losing run was 8 spins in a row (the worst of 20,000 was 22), and one session in ten put at least $20 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 −$2.69 to −$5.31 per $100, and the mean session from −$101 to −$208.

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.

-- Paroli: double after a win, bank the streak after three wins in a row.
-- 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 step

local function reset() step = 0 end

local function place(ctx)
  return { { type = "red", amount = UNIT * 2 ^ step } }
end

local function settle(result)
  if result.net > 0 then step = step + 1 else step = 0 end
  if step >= 3 then step = 0 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 Reverse Martingale system work?
• Does the 1-3-2-6 system work?
• Does the Contra D’Alembert system work?
• Study: Five strategies on the same seeds
• All 20 systems, compared on the same seeds