Heads-up Five Card Stud from 4th 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 4th-street Five Card Stud spots. The most a perfect opponent can take: under 0.008% of the pot, less than eight cents in a $1,000 pot.
- Spot ASeat 1K♥7♠7♦Seat 2A♣Q♦3♠0.0015%of the pot
- Spot BSeat 1A♠J♦9♣Seat 2Q♥Q♣5♦0.0078%of the pot
- Spot CSeat 110♠9♠8♦Seat 2K♦6♣2♥0.0026%of the pot
Exploitability after 1,000 iterations. Pots of 3 to 5 big bets at 4th street, three up-cards and one hole card each, a bet and three raises a street, both seats holding any hole card.
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’ hole cards and each 5th-street up-card, suits and all) and every betting line, 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 | Seat 1 shows | Seat 2 shows | Pot | First to act | Exploitable | In big bets | Seat 1’s value |
|---|---|---|---|---|---|---|---|
| A | K♥7♠7♦ | A♣Q♦3♠ | 4 | Seat 1 | 0.0015% | 0.000060 | +1.4362 |
| B | A♠J♦9♣ | Q♥Q♣5♦ | 3 | Seat 2 | 0.0078% | 0.000234 | -1.2695 |
| C | 10♠9♠8♦ | K♦6♣2♥ | 5 | Seat 2 | 0.0026% | 0.000128 | +0.0433 |
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 Five Card Stud from the rules: the betting, who acts first (the best hand showing), every deal of the hole cards and of each seat’s 5th-street up-card, card by card, the best five-card hand. Then it computes its own best responses.
Spot Exploitable: checker solver Seat 1’s value: checker solver A 0.0015% 0.0015% +1.4361885 +1.4361885 B 0.0078% 0.0078% -1.2695374 -1.2695374 C 0.0026% 0.0026% +0.0433035 +0.0433035 The three 4th-street spots above. All 186,310 decisions of each, after every 5th-street up-card for both seats, matched the rules as the checker rebuilt them.
- Matches
A separate reference solver
A reference solver written separately in Python solved Spot A. Run on the same update rule, HydraSolver’s exploitability matches it to every digit printed.
Iterations Reference HydraSolver 10 5.3586% 5.3586% 50 0.3582% 0.3582% Exploitability of Spot A, both solvers on the reference’s update rule.
- Matches
A spot with a textbook answer
5th street. Seat 1 shows A♥K♥Q♥J♥ and holds the ten of hearts in the hole, a royal flush, or the four of diamonds, ace-high, half and half. Seat 2 shows K♦9♠5♣2♣ and holds a nine in the hole, a pair of nines. 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: 4th street to the showdown
Heads-up, the last two of Five Card Stud’s four betting rounds, each 5th-street up-card dealt card by card. Every hand, every card. The exploitability above is computed, not estimated. When a hand reaches 4th street, HydraSolver switches to this solve.
Estimated: 2nd and 3rd street
The whole hand from 2nd street is solved by self-play over sampled deals, with similar hands grouped together. Its accuracy is measured statistically (on 2nd 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 Five Card Stud
Whole hands from 2nd street
Heads-up or three-handed, with your own door cards, ante, bring-in and limits. Walk the hand street by street, dealing each up-card yourself or at random.
Exact play from 4th street
When a heads-up hand reaches 4th street, the rest of it is solved exactly for the line you are on, and keeps sharpening while you read it.
Every hand, two ways
Hand classes the way players name them (a pair, two big cards, a draw), and hand by hand, each with its EV.
Your own spots
Set up any heads-up 4th- or 5th-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). Five Card Stud 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.