The last round of heads-up A-5 Single Draw, solved exactly
Any solver can print a strategy. The real question is whether it’s right. In two-player poker there is a hard test: give the strategy to a perfect opponent who knows it, let them find the best counter-play, and count what they win. We ran that test on the last round of three A-5 Single Draw hands, between the hands each player brings to it. The most a perfect opponent can take: under 0.003% of the pot.
- Spot ARaise, call. Each draws two.Pot 2 big bets0.0019%of the pot
- Spot BCall, check. The big blind draws two, the button three.Pot 1 big bet0.0027%of the pot
- Spot CRaise, re-raise, call. The big blind draws one, the button two.Pot 3 big bets0.0014%of the pot
Exploitability after 1,000 iterations, a bet and three raises. Each seat’s hands are the ones HydraSolver’s own solve of the hand brings to the last round on that line, up to 5,000 drawn by weight.
What “solved” means
A strategy pair is solved when neither player can win more by changing their play. How far a strategy is from that has a name: exploitability. Let an opponent see the strategy, play the best possible response to it, and count what they win; average that over both seats. Zero means nobody can beat it.
We compute that number exactly, not by sampling. The best response is worked out over every pair of hands the two players can hold (two hands meet only when they share no card) and every betting line of the round, with no hands lumped together. It’s the same yardstick researchers used when they declared heads-up limit hold’em “essentially solved” (Bowling et al., Science, 2015).
The result
| Spot | Button’s hands | Big blind’s | Pot | First to act | Exploitable | In big bets | Button’s value |
|---|---|---|---|---|---|---|---|
| A | 5,000 | 5,000 | 2 | Big blind | 0.0019% | 0.000038 | +0.1059 |
| B | 5,000 | 5,000 | 1 | Big blind | 0.0027% | 0.000027 | -0.0307 |
| C | 5,000 | 5,000 | 3 | Big blind | 0.0014% | 0.000043 | -0.4953 |
Values in big bets, the button’s side. The pot at the start of the last round.
Checking the check
A number from a solver about its own strategy is only as good as the code that computed it. So we checked it three separate ways.
- Matches
A second program, written separately
An independent checker in a different language reads only the saved strategy and each seat’s hands. It rebuilds the last round from the rules: the betting, two hands meeting only when they share no card, and A-5 lowball (the lowest hand wins: aces low, straights and flushes do not count, pairs do). Then it computes its own best responses.
Spot Exploitable: checker solver Button’s value: checker solver A 0.0019% 0.0019% +0.1058914 +0.1058914 B 0.0027% 0.0027% -0.0307470 -0.0307470 C 0.0014% 0.0014% -0.4953014 -0.4953014 The three spots above. All 10 decisions of each matched the rules as the checker rebuilt them.
- Matches
Every pair, counted one at a time
Which hands can meet is counted in one pass over each range. On random ranges in 2-7, A-5, Badugi and Badacey, every hand’s share of the pot and the weight of the hands it can meet match counting every pair of hands one at a time, to the ninth decimal.
- Matches
A spot with a textbook answer
The last round, pot two big bets. The big blind holds the wheel, A-2-3-4-5, or a pair of kings, half and half: A♣2♦3♥4♠5♣ or K♣K♦Q♥J♠10♦. The button holds a nine-low, which beats the kings and loses to the wheel: 9♠8♥7♦6♣4♥. One bet, no raise. Theory: bluff a third of the air, call twice in three, worth exactly 1/3 of a bet to the big blind.
Figure Theory HydraSolver Air that bluffs 33.333% 33.330% The middle hand calls 66.667% 66.674% Big blind’s value 0.333333 0.333333
What’s exact, and what isn’t
Exact: the last betting round
Heads-up, after the draw: every hand each seat brings to it, every card. The exploitability above is computed, not estimated. When a hand on screen reaches the last round, HydraSolver switches to this solve.
Estimated: the deal and the draw
The whole hand is solved by self-play over sampled deals, with similar hands grouped together, and the draw on the line on screen is worked out hand by hand; the hands each seat brings to the last round come from that. That’s measured in play, not proven.
What HydraSolver gives you for A-5 Single Draw
Whole hands from the deal
Heads-up, with your own blinds, at fixed limit, or at pot limit or no limit with your own stack and bet sizes. Walk the hand round by round, with any hand you type.
The draw worked out hand by hand
At the draw, each hand’s draw is the one that wins it most against the other seat’s hands on the line.
The last round solved exactly
After the draw, the betting is solved exactly for the line you are on, hand by hand with every card counted, at every limit.
Saved, on your machine
Hands save and reopen like every other HydraSolver solve. It all runs on your own computer.
More proofs
Measured with the solver in HydraSolver 1.1.98 (October 18, 2026). A-5 Single Draw is in beta in that version. Exploitability is the average of what a best response wins against each seat’s strategy, as a share of the pot at the start of the spot.