The last round of heads-up A-5 Triple 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 Triple Draw hands, between the hands each player brings to it. The most a perfect opponent can take: under 0.002% of the pot.
- Spot ARaise, call. Each draws two. Check, bet, call. Each draws one. Check, bet, call. The big blind draws one, the button stands pat.Pot 5 big bets0.00073%of the pot
- Spot BCall, check. The big blind draws two, the button three. Check, check. The big blind draws one, the button two. Bet, call. Each draws one.Pot 3 big bets0.0015%of the pot
- Spot CRaise, re-raise, call. The big blind draws one, the button two. Bet, call. Each draws one. Check, check. The big blind stands pat, the button draws one.Pot 4 big bets0.0018%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 | 2,000 | 5,000 | 5 | Big blind | 0.00073% | 0.000036 | +1.3986 |
| B | 5,000 | 5,000 | 3 | Big blind | 0.0015% | 0.000045 | +0.1553 |
| C | 5,000 | 2,000 | 4 | Big blind | 0.0018% | 0.000073 | -0.7212 |
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 0.00073% 0.00073% +1.3986366 +1.3986366 B 0.0015% 0.0015% +0.1552585 +0.1552585 C 0.0018% 0.0018% -0.7212182 -0.7212182 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 five 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 sixth of the air, call five times in six, worth exactly 5/12 of a bet to the big blind.
Figure Theory HydraSolver Air that bluffs 16.667% 16.669% The middle hand calls 83.333% 83.338% Big blind’s value 0.416667 0.416667
What’s exact, and what isn’t
Exact: the last betting round
Heads-up, after the third 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 to the third draw
The whole hand is solved by self-play over sampled deals, with similar hands grouped together. From the betting after the second draw the line on screen is solved again on its own, and each draw before it 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 Triple Draw
Whole hands from the deal
Heads-up, with your own blinds and limits. Walk the hand round by round and draw by draw, with any hand you type.
Each line’s end solved on its own
From the betting after the second draw, the line you are on is solved again for the hands each seat brings to it.
Every draw worked out hand by hand
At each 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 third draw, the betting is solved exactly for the line you are on, hand by hand with every card counted.
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 Triple 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.