Kuhn poker, and why it is worth solving
Kuhn poker is the smallest game that is still genuinely poker. Three cards, one betting round, two players — and yet it already contains bluffing, bluff-catching, and the need to randomise. Small enough that the perfect strategy is known on paper; rich enough that the perfect strategy is nothing like what your intuition suggests.
The rules
A complete hand takes about ten seconds.
The only thing you can lose is chips you put in. You ante 1 no matter what. If you bet and get called you risk 1 more. So every hand settles at ±1 chip (a check-down or a fold) or ±2 chips (a bet that got called).
Every way a hand can go
Nine possible betting sequences. That is the entire game — nothing is being summarised here.
What makes it hard
You never see the other card. That single fact breaks every technique that works on chess or tic-tac-toe. In chess you can look at the board and ask “what is the best move here?” In poker there is no “here” — you are in one of several situations at once and have to pick one action that covers all of them.
An “information set” is everything you know when it is your turn: your own card, plus the betting so far. If you hold the Queen and your opponent has bet, you are in the same information set whether they hold the Jack or the King — so you must choose one plan that handles both. There are 12 such situations in the whole game. Solving Kuhn poker means finding the right answer to all 12.
And the right answer is usually not a single action. It is a frequency — bet this often, call that often. A player who always does the same thing in the same spot is readable, and a readable poker player is a losing poker player. That is the deep reason the solution has to be random.
You are mathematically required to bluff
Not as a trick or a “mind game” — as arithmetic.
Suppose you decide, sensibly, to bet only the King. You are never “caught” bluffing and you never throw chips away. Now think about what your opponent sees: you bet, therefore you have the King. They fold everything else, instantly, forever. Your King never gets paid. And every time you hold the Jack, you check and lose at showdown.
Betting the Jack changes that. It is a losing card — it wins no showdown ever — so its only path to winning a pot is to make a better hand fold. Against the “bet only the King” opponent, the numbers come out:
Check → they check back or bet you off it = −1.00 chips
Bet → they fold the Queen; the King calls = −0.50 chips
Bluffing the worst card in the deck cuts your loss in half. This is the single most counterintuitive fact in the game, and it is why every solved poker strategy bluffs: the bluff is not a gamble bolted onto a solid strategy, it is load-bearing structure.
The 3:1 rule. At equilibrium Player 1 bets the King exactly three times as often as the Jack. Why exactly three? Because that is the ratio that makes a Queen facing your bet completely indifferent between calling and folding:
Given a bet, you hold J with prob 1/4, K with prob 3/4
Call → (1/4)(+2) + (3/4)(−2) = −1.00
Fold → give up the ante = −1.00
Dead even. There is no read to make, no tell to find, no adjustment that helps them. That is what a solved strategy does — it does not outguess the opponent, it removes the guess.
And you are required to call with a hand that usually loses
The mirror image. When you hold the Queen facing a bet, you beat only the Jack. Folding feels right. But if you always fold the Queen, your opponent can bet every Jack as a bluff and print chips. So you have to call sometimes — and there is exactly one correct frequency:
Bluff succeeds → win the 2-chip pot = +1
Bluff gets called → lose ante + bet = −2
Break-even when: 3·P(fold) − 2 = −1 → P(fold) = 1/3
Which requires the Queen to call exactly 1/3 of the time.
Call less than 1/3 and bluffing becomes profitable against you. Call more and you bleed chips to genuine Kings. One number, and it is not a matter of taste.
The solved strategy
Computed live by this app's CFR solver — not hard-coded — then checked against the closed-form answer Kuhn published in 1950.
Read Player 2's rows carefully. Bet the Jack 1/3 of the time when checked to. Call the Queen 1/3 of the time when bet into. Those exact thirds are the numbers derived above — the solver was never told them. It found them by playing itself several hundred thousand times and regretting its mistakes.