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Kuhn Poker Rules

Kuhn poker is a simplified two-player poker game played with just three cards: a king, a queen and a jack. Each player antes one chip and gets one card, and there is a single round in which each player may bet one chip. The higher card wins at showdown. It was devised as a model for studying bluffing mathematically.

Players
2
Deck
3 cards: king, queen, jack
Cards
1 per player
Bets
Ante 1, bet 1

Article 1Players, deck and objective

1.1

Kuhn poker was devised by the mathematician Harold W. Kuhn in 1950 as a simple two-player, zero-sum game of imperfect information. It is used to study poker strategy and game theory rather than played for serious stakes.

1.2

The game is for exactly two players and uses a deck of three cards: a king, a queen and a jack. The king is highest and the jack lowest. Suits do not matter.

1.3

The object is to win chips from the opponent, either at showdown with the higher card or by betting when the opponent folds.

Article 2Antes and the deal

2.1

Each player antes one chip, making a pot of two.

2.2

Each player is dealt one card face down. The third card is set aside unseen and takes no part in the hand.

Article 3The betting

There is one betting round. A bet is always exactly one chip, and there are no raises.

3.1

The first player may check or bet one chip.

3.2

If the first player bets, the second player may fold, losing the ante, or call by adding one chip, which leads to a showdown for a pot of four.

3.3

If the first player checks, the second player may check, which leads to a showdown for a pot of two, or bet one chip.

3.4

If the first player checks and the second player bets, the first player may fold or call. A call leads to a showdown for a pot of four.

SequenceResult
Check, checkShowdown; the winner gains 1 chip
Bet, foldFirst player wins; gains 1 chip
Bet, callShowdown; the winner gains 2 chips
Check, bet, foldSecond player wins; gains 1 chip
Check, bet, callShowdown; the winner gains 2 chips

Article 4Showdown

4.1

At showdown, the player with the higher card wins the pot. Ties are impossible.

Example

Checking and folding

The first player holds J♠ and the second player holds K♠. The Q♠ is set aside.

The first player checks. The second player bets one chip. Holding the lowest card, the first player folds and loses the one-chip ante.

Example

A called bluff

The first player holds J♠ and bets. The second player holds Q♠ and calls.

The queen beats the jack. The second player wins the pot of four, a gain of two chips.

Article 5Strategy notes

Kuhn poker has been solved. Its equilibrium strategies show that bluffing is part of optimal play, not a mistake.

  • The second player's equilibrium strategy is unique: always bet or call with the king; with the queen, check when possible and call a bet one time in three; with the jack, never call, but bet one time in three after a check.
  • The first player has a family of equilibrium strategies governed by a value α between 0 and 1/3: bet the jack with probability α, bet the king with probability 3α, always check the queen and call a bet with it with probability α + 1/3.
  • Under equilibrium play, the first player loses on average 1/18 of a chip per hand, so the second position is an advantage.

Article 6Frequently asked questions

How many cards are used in Kuhn poker?

Three: a king, a queen and a jack. Each player gets one, and the third is set aside unseen.

Who invented Kuhn poker?

The mathematician Harold W. Kuhn, in 1950, as a simple model for studying poker with game theory.

Should you bluff in Kuhn poker?

Yes, sometimes. The equilibrium strategy includes betting the jack, the worst card, a fixed fraction of the time.

§
Final authority

Where any doubt remains in the application of this document, Rob is right.

Hand rankings, high to low: royal flush · straight flush · four of a kind · full house · flush · straight · three of a kind · two pair · one pair · high card. Suits never break ties.

Kuhn Poker Rules · OPR-405 · officialpokerrules.com/kuhn-poker/

OPR-405Uncontrolled when printed