Heads-up High-Low Chicago from 6th street, 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 over every card that can come, and count what they win. We ran that test on three 6th-street High-Low Chicago spots. The most a perfect opponent can take: under 0.001% of the pot, less than a cent in a $1,000 pot.
- Spot ASeat 19♥9♦K♣4♣Seat 2J♥8♥5♥Q♠0.00065%of the pot
- Spot BSeat 1K♠Q♦8♣2♥Seat 210♥10♣6♦4♠0.00032%of the pot
- Spot CSeat 1A♦J♣9♠5♠Seat 27♥7♦A♣3♣0.00042%of the pot
Exploitability after 1,000 iterations. Pots of 4 to 6 big bets at 6th street, a bet and three raises a street, both seats holding any two down cards.
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 way the unseen cards can be dealt (both players’ down cards and each 7th-street card, suits and all) and every betting line, with no hands lumped together, both halves of every pot. 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 | Seat 1 shows | Seat 2 shows | Pot | First to act | Exploitable | In big bets | Seat 1’s value |
|---|---|---|---|---|---|---|---|
| A | 9♥9♦K♣4♣ | J♥8♥5♥Q♠ | 5 | Seat 1 | 0.00065% | 0.000032 | +0.5402 |
| B | K♠Q♦8♣2♥ | 10♥10♣6♦4♠ | 4 | Seat 2 | 0.00032% | 0.000013 | -0.7510 |
| C | A♦J♣9♠5♠ | 7♥7♦A♣3♣ | 6 | Seat 2 | 0.00042% | 0.000025 | -0.7727 |
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. It rebuilds High-Low Chicago from the rules: the betting, who acts first, every deal of the unseen cards card by card, the best five-card hand and the highest spade in the hole, half the pot to each, the high taking all of it when nobody has a spade in the hole. Then it computes its own best responses.
Spot Exploitable: checker solver Seat 1’s value: checker solver A 0.00065% 0.00065% +0.5402093 +0.5402093 B 0.00032% 0.00032% -0.7510023 -0.7510023 C 0.00042% 0.00042% -0.7726669 -0.7726669 The three 6th-street spots above. All 100 decisions of each matched the rules as the checker rebuilt them.
- Matches
A separate reference solver
A reference solver written separately in Python valued a 6th-street spot in each of four stud games, Spot A among them. HydraSolver agrees to within a ten-thousandth of a big bet.
Game Reference HydraSolver Seven Card Stud +1.2134 +1.2134 Razz -1.3220 -1.3220 Stud Hi-Lo -0.5710 -0.5710 High-Low Chicago +0.5403 +0.5402 Seat 1’s value in big bets. Spot A is the reference’s High-Low Chicago spot.
- Matches
A spot with a textbook answer
7th street. Seat 1 shows A♥K♥Q♥4♥ and holds a royal flush with the queen or jack of spades in the hole, so it wins the whole pot, or ace-high with only the three of spades in the hole, half and half. Seat 2 shows K♦9♠5♣2♣ and holds a pair of nines with the six or eight of spades in the hole: it takes the whole pot from the ace-high and loses all of it to the royal. Pot 5, one bet, no raise. Theory: bluff a sixth of the air, call five times in six, worth exactly 5/12 of a bet to Seat 1.
Figure Theory HydraSolver Air that bluffs 16.667% 16.670% Nines calls 83.333% 83.336% Seat 1’s value 0.416667 0.416667
What’s exact, and what isn’t
Exact: 6th street to the showdown
Heads-up, the last two of High-Low Chicago’s five betting rounds. Every hand, every card, both halves of every pot. The exploitability above is computed, not estimated. When a hand reaches 6th street, HydraSolver switches to this solve.
Estimated: 3rd to 5th street
The whole hand from 3rd street is solved by self-play over sampled deals, with similar hands grouped together. Its accuracy is measured statistically (on 3rd street it shows no loss it can detect after 40 million hands, to within about one big bet per 100 hands). That’s an estimate, not a proof. Three-handed stud isn’t covered here.
What HydraSolver gives you for High-Low Chicago
Whole hands from 3rd street
Heads-up or three-handed, with your own doors, ante, bring-in and limits, and the open-pair big bet on 4th street. Walk the hand street by street, dealing each up-card yourself or at random.
Exact play from 6th street
When a heads-up hand reaches 6th street, the rest of it is solved exactly for the line you are on, and keeps sharpening while you read it.
Every hand, three ways
A down-card chart, hand classes the way players name them (trips, two pair, flush draws), the ace or king of spades in the hole on its own, and hand by hand, each with its EV.
Your own spots
Set up any heads-up 6th- or 7th-street spot: the up-cards, the dead cards, the pot and each seat’s range. Its exploitability and both seats’ EVs show as it solves.
Saved, on your machine
Hands and spots save and reopen like every other HydraSolver solve. It all runs on your own computer.
More proofs
Measured with the solver in HydraSolver 1.1.96 (October 14, 2026). High-Low Chicago 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.