Kuhn Poker

Rules

The deck has three cards: Jack, Queen, King. Both players ante one chip and get one card. The higher card wins a showdown.

The first player checks or bets one chip.

• After a bet, the second player calls for a showdown or folds and gives up the antes.
• After a check, the second player checks for a showdown or bets. Facing that bet, the first player calls or folds.

A hand is worth one chip if nobody calls and two chips if a bet gets called. Seats alternate every hand, and a coin flip decides who opens the first one.

Each player starts with ten chips. A hand can cost two, so the game ends as soon as either player has fewer than two chips left.

Why it is worth a few hands

Harold Kuhn solved this game by hand in 1950, and it is the smallest poker variant whose optimal strategy still bluffs and traps. Played perfectly by both sides, the first player loses 1/18 of a chip per hand, so position is worth something even here. Seats alternate, so with perfect play on both sides the long-run average is zero.

The opponent plays the equilibrium strategy below. Against it there is no way to win in the long run, only ways to lose faster, so the interesting question is how far your instincts drift from it. Turn on the advice to compare.

Equilibrium strategy

The second player’s strategy is unique. The first player has a family of them, set by a bluffing rate α anywhere from 0 to 1/3. The opponent style above picks it: passive is α = 0, balanced is 1/6, aggressive is 1/3.

Swipe or scroll sideways to see all columns.

Bets with a Jack are pure bluffs, and checks with a King, when α is below 1/3, are traps. The Queen never bets, because a bet with the Queen only gets called by the King.