Can collusion beat Ultimate Texas Hold'em?
Two articles appear to reach very different answers. An exact solver and 100,000 six-player hands show which numbers hold up.
No. Even perfect six-player collusion still loses about 0.87% of the Ante. Both articles reach the right practical conclusion, but a 2.5% player edge is not supported. It is well within the noise of the 500-hand sample behind it.
The disagreement
Ultimate Texas Hold'em (UTH) is a casino game played against the dealer. Players at a table can each see only their own two cards, unless they share them. A full six-spot table therefore gives a collusion team 12 known cards, including the player's own, and the question is how much that information is worth.
A 2010 Discount Gambling simulation found that practical collusion cuts the six-player house edge from 2.2% to 1.6%. A July 2024 Advanced Advantage Play article put the collusion edge at about 2.5%, using all 12 cards but only 500 hands.
Part of the apparent conflict is about measurement. Discount Gambling's strategy tables show gains of 20% or more, but only for specific situations, such as raising K5o when no one else holds a king or five. That situation occurs on only about 0.2% of hands, so large situational gains become small overall ones.
The line shows the approximate 95% range of a raw 500-hand average, using the standard deviation measured in this check. It assumes the 2.5% is a raw average, which the article does not state.
The results
| Six players, 100,000 hands | Player EV | Gain over basic strategy |
|---|---|---|
| No collusion (exact) | −2.185% | n/a |
| Post-flop collusion only | −1.96% | +0.225% ± 0.002 |
| Discount Gambling's table rules pre-flop, no collusion after | −1.66% | +0.53% ± 0.02 |
| Perfect pre-flop collusion, no collusion after | −1.10% | +1.08% ± 0.03 |
| Perfect collusion on all three streets | −0.87% | +1.31% ± 0.03 |
EV is expected value. Figures are percentages of the Ante, ranges are 95% margins, and the Trips side bet is excluded.
Most of the gain comes from the pre-flop choice between a 4x raise and a check. Using collusion only after the flop, with basic strategy before it, adds just 0.225%.
At a five-player table, perfect collusion is weaker still: −1.26% ± 0.06, a gain of +0.93%, from 11,797 hands.
Advanced Advantage Play: sound calculations, noisy headline
The article's detailed calculations hold up. The solver matched both worked examples to two decimal places:
| Example | Check EV | Raise 4x EV |
|---|---|---|
| J♠T♠ with four pair outs visible | +33.54% | −12.17% |
| K♣3♥ with no pair outs visible | +7.96% | +28.86% |
Its pre-flop statistics are also close. The solver found 45.5 deviations from basic strategy per 500 hands, against the article's 47, with an average gain of 11.9%, against 11.2%.
The 2.5% is the outlier. It implies a gain of about 4.7 points, while the solver finds 1.31. The likely cause is sample size: a raw 500-hand average has a standard deviation of about 3.7%, and about 18% of such samples would show +2.5% or better. The author warned that the sample was too small and the results very approximate, and suggested 100,000 hands.
The article's other conclusions stand. Five-player UTH shows no collusion edge, and collusion is not a serious threat to the game.
Discount Gambling: a good overall estimate
The reported gain of about 0.6% looks right. Simple raise-or-check rules taken from its tables gain 0.53%. That rises to 0.75% if the team also colludes perfectly after the flop, and its 0.6% includes kicker strategies, so it sits within that range.
Its strategy tables give the right decisions. All 49 values tested, at two, four and six players, gave the same raise-or-check answer. The sizes usually run slightly high, however, often by 0.5–3 points:
| Six players: raise 4x minus check | Article | This check |
|---|---|---|
| 44, no outs seen | +41% | +34.2% to +35.2% |
| JTo, no outs seen | +33% | +30.7% to +31.4% |
| Q8o, no outs seen | +26% | +23.5% to +24.0% |
| 33, no outs seen | +24% | +20.0% to +20.9% |
| K5o, no outs seen | +22% | +22.4% to +23.0% |
| K3o, no outs seen | +13.1% | +12.4% to +12.9% |
| K5o, one king seen | +7.5% | +7.1% to +7.7% |
| 22, no outs seen | +4.7% | +3.2% to +3.6% |
| 22, one out seen | −32% | −33.3% to −32.9% |
The article does not state its post-flop assumptions, so each range runs from post-flop play using all known cards to post-flop play using none of the collusion information. Sampling error is typically 0.3–0.6 points per value.
How this was checked
The solver works through every remaining flop, turn, river and dealer hand for each deal, using 4x, 3x and check before the flop, 2x or check on the flop, and 1x or fold on the river. It did not use sampling inside a hand, only in choosing which deals to test.
The rules used are the standard ones: Ante and Blind equal, the dealer qualifies with a pair or better, and the Blind pays 500:1 on a royal flush, 50:1 on a straight flush, 10:1 on four of a kind, 3:1 on a full house, 3:2 on a flush and 1:1 on a straight.
Before any collusion figures were trusted, it had to pass these tests:
- Hand frequencies: all categories matched the known counts across all 133,784,560 seven-card hands.
- House edge: the exact no-collusion figure came out at 2.185%, matching the published value.
- Independent recount: the fast method's 4x raise EV matched a slow, separate enumeration of 391,514,760 outcomes per deal on three random deals, to 10 decimal places.
- Published examples: it reproduced both Advanced Advantage Play examples and Discount Gambling's +2.3% no-information value for Q8o.
A raw average is dominated by which hole cards happen to be dealt. To remove most of that noise, the solver also played each deal with the no-collusion strategy, measured the gain from collusion on those same cards, and added that gain to the exact −2.185%. This cut the noise by a factor of about 17. As a cross-check, the raw six-player average was −0.96% ± 0.50, consistent with −0.87%.
Limits of this check
- The −0.87% assumes an exact calculation on every decision using all 12 cards, which no team could do at a live table. It is a ceiling, and real results would be worse.
- Comps and other extras are excluded, and could change the result for a particular player.
- The 0.53% rests on one translation of Discount Gambling's tables into rules. Where the tables are silent, it mostly assumes checking marginal hands when two outs are seen, and it leaves unlisted suited hands at basic strategy.
- Only the two-, four- and six-player columns of the tables were tested, and the "low out" and "high and low out" rows were not.
- Advanced Advantage Play's spreadsheet was not reviewed, so the sampling explanation is the most likely cause rather than a confirmed one.
Sources
- Stephen How, "Collusion Performance for Ultimate Texas Hold'Em", Discount Gambling, June 14, 2010.
- Stephen How, "Practical Collusion for Ultimate Texas Hold'em", Discount Gambling, January 15, 2010.
- "Ultimate Texas Hold'em: Collusion", Advanced Advantage Play, July 11, 2024.