Workbook C
Solver baselines
Limit draw games have had far less solver attention than hold'em. We ran our own. This page gives every number that came out, and is just as careful about what the numbers cannot tell you.
1What was solved
A simplified, two-player version of each limit triple-draw game, solved with CFR+, a standard self-play method that moves a pair of strategies toward the point where neither can gain by changing. Three simplifications matter:
- Hand classes, not hands. The solver groups hands into classes and plays every hand in a class the same way. The real game has finer distinctions than that.
- No card removal between the two players. In the real game the cards you hold cannot also be in your opponent's hand. The simplified game ignores this.
- Heads-up only. The six-handed figures below are an approximation built from heads-up solves, not a solve of a six-player game.
How to say it. These are baselines from our solver for a simplified game. They are not a solution to triple draw, and nothing on this site should be read as one. Where the simplified game and the real game differ, the baseline will be wrong, and we do not yet know by how much.
2How well the simplified game was solved
| Game | Exploitability |
|---|---|
| 2-7 triple draw | 0.0018 |
| Ace-to-five triple draw | 0.0018 |
| Badugi | 0.0027 |
Exploitability is how much a perfect opponent could win per hand against the strategy. These are small, which means the simplified game was solved closely. It says nothing about how close the simplified game is to the real one. A perfect answer to an easier question is still an answer to an easier question.
3Heads-up 2-7 triple draw
| Spot | What the baseline does |
|---|---|
| Button, first to act | Raises about 85% of hands |
| Big blind facing that raise | Folds 16%, calls 62%, re-raises 22% |
| Pat sevens and eights | Kept |
| Rough nine (9-8), opponent drew one, bet, then stood pat | Broken about 98% of the time |
| Smooth nine, same opponent but he only check-called | Kept 94% of the time |
A known weakness. We suspect the 85% opening figure is too wide, and that the missing card removal is the reason.
4Six-handed: who opens how often
| Game | UTG | HJ | CO | BTN | SB |
|---|---|---|---|---|---|
| 2-7 triple draw | 21% | 22% | 30% | 43% | 70% |
| Ace-to-five triple draw | 19% | 25% | 33% | 50% | 58% |
| Badugi | 18% | 21% | 23% | 38% | 50% |
These figures are an approximation built from heads-up proxy solves, not a solve of a six-player game. Use the shape (tight up front, much looser on the button and in the small blind) with more confidence than any single figure.
5Exact odds quoted with the solver work
These are not solver output. They are plain counting, and our own odds script reproduces each one.
| Deuce-to-seven | 1 draw left | 2 left | 3 left |
|---|---|---|---|
| 2-3-4-7 drawing one makes an eight or better | 25.0% | 44.1% | 58.7% |
| A pat 9-8-6-4-2 beats a 2-3-4-7 draw | 69.8% | 48.2% | 32.9% |
The first row counts 12 helpful cards among 48: the four cards you keep are removed and the card you throw away is not counted as seen. If you count your discard as a known dead card the figures become 25.5%, 45.0% and 59.6%, which are the ones used on the game pages. The second row counts 13 helpful cards among the 43 left when both hands are known.
6How each player type loses money
Pending · not yet published
A fuller report is being computed on how each type of player loses money. It is not finished, and nothing is shown here until it is. This section is its reserved place. No figures on this site anticipate it.
7How to use a baseline
- As a first answer to "roughly how often", to be checked against experience and against the published advice on each game page.
- For its patterns more than its percentages. That the same nine is kept against one betting line and broken against another is a durable idea; "98%" is a property of the simplified game.
- Not as a chart to memorise, and not for the split-pot games, single draw, or five-card draw, none of which it covers.