What breaks it: American wheel, long sessions and table minimums
The gain survives other seeds and a bigger bankroll, but long sessions, $25 table minimums and, for wide layouts, the double-zero wheel break the ladders; only the cash-out version shrugs off length and the wheel.
Chapter 8 of the spinner roulette research (chapters 0 to 9): What breaks it.
We re-ran the survivors under each change. Other seeds moved the typical session by at most $19. A $10,000 bankroll and the la partage rule changed nothing. The American wheel’s extra zero costs 3 to 8 points of chance of ending ahead; b3 and c9 stay ahead, the widening a1 drops to −$1,978. At 1,000 spins every ladder without a cash-out ends negative.
The c9 cash-out shrugs most of this off: every session stops by spin 250, so length does not touch it, and on the American wheel it keeps +$286 and 88.7 %. At $25 table minimums nothing works: comparable ladders lost the ladder in 90 % to 100 % of sessions. Every average is negative.
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The numbers
| Strategy | European, 500 spins | American wheel | 250 spins | 1,000 spins | $10,000 bankroll |
|---|---|---|---|---|---|
| Flat $25 on red | -$350 / -$338.21 | -$650 / -$653.14 | -$150 / -$170.85 | -$650 / -$675.80 | -$350 / -$338.21 |
| b3: two streets, stacked | +$1,679 / -$320.90 | +$1,594 / -$520.74 | +$862 / -$166.99 | -$1,947 / -$474.46 | +$1,679 / -$320.90 |
| c9: streets to 12, stacked | +$1,540 / -$295.45 | +$1,497 / -$535.24 | +$777 / -$156.42 | -$2,126.50 / -$472.93 | +$1,540 / -$295.45 |
| a1: streets, widening | +$1,494 / -$330.93 | -$1,978 / -$699.60 | +$775 / -$190.84 | -$1,568 / -$538.08 | +$1,494 / -$330.93 |
| Strategy | Typical session | Chance of ending ahead | Average | Worst 5 % |
|---|---|---|---|---|
| c9 + cash-out, European, 500 spins | +$286 | 90.3% | -$46.25 | -$3,210.44 |
| c9 + cash-out, American wheel | +$286 | 88.7% | -$100.32 | -$3,219.87 |
| c9 + cash-out, 250 spins | +$286 | 90.3% | -$46.25 | -$3,210.44 |
| c9 + cash-out, 1,000 spins | +$286 | 90.3% | -$46.25 | -$3,210.44 |
| c9 + cash-out, American wheel, 1,000 spins | +$286 | 88.7% | -$100.32 | -$3,219.87 |
| a1 + $226 cash-out, full ladder | +$232 | 91.7% | -$63.55 | -$3,380.37 |
| a1 without its last level + $226 cash-out | +$230 | 72.4% | -$38.08 | -$840.80 |
| a1 without its last level, no cash-out | -$360 | 30.6% | -$124.31 | -$840.80 |
Takeaways
- Plan the length: without a cash-out these ladders are built for about 500 spins.
- Play on $1 minimums and a single-zero wheel; the double zero hurts wide layouts most.
- Widening needs a shorter ladder, not a stop-loss: dropping a1’s last level with a $226 cash-out cuts the worst 5 % from −$3,380 to −$841, for 19 points of chance (91.7 % → 72.4 %).
The math
American wheel: p(k) = c(k)/38 and the edge is 2/38 = 5.26 %; realized −5.43 % (b3), −4.91 % (c9), −5.47 % (a1), −5.23 % (flat red). q grows by Π (1 − c(k)/38) / (1 − c(k)/37) over the levels: b3 0.00591 → 0.00685, predicted P(no cap) 0.58, measured 0.585; a1 predicted 0.49, measured 0.473, below ½, so its median flips. The c9 cash-out: q_SL 0.00510 × 1.181 = 0.00602, (1 − q)^20 = 0.886, measured 0.887; mean −$100.32 (−4.75 % of $2,113).
Length: P(no cap at S) = e^(−k·p·S/B) falls below ½ at S½ = ln 2 × B/(k·p); at 1,000 spins the medians stay ahead until 690, 860, 570 (b3, c9, a1) against n½ 723, 874, 577. La partage halves the edge on even money only: flat red −1.36 %, mean −$169.82. With onCap stop the cap loss ($3,444 to $4,919) stays under $5,000, so extra bankroll is never used. Default table ($25 minimums): a9, b11, c6 kept 0.3 %, 9.8 % and 0 % of 2,000 sessions uncapped.
edge (American) = −2/38 = −5.26 %
q(American) = q × Π (1 − c(k)/38) / (1 − c(k)/37)
S½ = ln 2 × B / (k·p)