kuhn-poker

Table of Contents

1. Overview
2. Game Rules
3. Game Tree Structure
4. Information Sets
5. Nash Equilibrium
6. Why Kuhn Poker Matters
7. CFR Results
8. Sources

Overview

Kuhn Poker is a simplified form of poker developed by mathematician Harold W. Kuhn as a tractable model for studying zero-sum two-player imperfect-information games. Despite its simplicity, it captures essential poker dynamics including bluffing, slow-playing, and strategic deception.

Classification: Two-player, zero-sum, finite game with imperfect information and chance moves.

Game Rules

Setup

• Deck: 3 cards (King > Queen > Jack)
• Players: 2
• Ante: Each player antes 1 chip

Deal

One card dealt to each player (hidden from opponent). One card remains unseen.

Betting Round

Player 1 acts first:

• Check (pass): No additional bet
• Bet: Add 1 chip to pot

Player 2 responds:

• If Player 1 checked:
  • Check: Showdown
  • Bet: Add 1 chip
• If Player 1 bet:
  • Fold: Player 1 wins pot
  • Call: Add 1 chip, showdown

If Player 2 bet after Player 1 check:

• Player 1 can Fold or Call

Showdown

Higher card wins the pot.

Payoff Structure

Game Tree Structure

                    [Chance: Deal Cards]
                           |
            /--------------+--------------\
           /               |               \
        P1:J            P1:Q             P1:K
          |               |                |
       /-----\         /-----\          /-----\
    Check   Bet     Check   Bet      Check   Bet
      |       |       |       |        |       |
    P2:?    P2:?    P2:?    P2:?     P2:?    P2:?
      |       |       |       |        |       |
    /---\   /---\   /---\   /---\    /---\   /---\
   C   B   F   C   C   B   F   C    C   B   F   C
   |   |   |   |   |   |   |   |    |   |   |   |
  SD  P1  +1 SD   SD  P1  +1 SD    SD  P1  +1 SD
      |               |                 |
    /---\           /---\             /---\
   F   C           F   C             F   C
   |   |           |   |             |   |
  -1  SD          -1  SD            -1  SD

Legend: C=Check, B=Bet, F=Fold, SD=Showdown

Information Sets

An information set contains all game states indistinguishable to a player. In Kuhn Poker:

Player 1 Information Sets (6 total):

Player 2 Information Sets (6 total):

Total: 12 information sets in the game.

Nash Equilibrium

Game Value

At Nash equilibrium, Player 1 expects to lose -1/18 per hand (≈ -0.0556).

This means Player 2 has a slight positional advantage from acting second.

Equilibrium Properties

Kuhn Poker has infinitely many Nash equilibria, but all share the same game value of -1/18.

There is no pure strategy equilibrium - optimal play requires randomization (mixed strategies).

Equilibrium Strategy Ranges

Player 1 with Jack (weakest card):

• First action: Check ~79%, Bet ~21%
• After Check-Bet: Always Fold

The ~21% betting with Jack is a bluff - betting with a weak hand to potentially win without showdown.

Player 1 with King (strongest card):

• First action: Mix of Check and Bet
• After Check-Bet: Always Call

Checking with King is slow-playing - concealing strength to extract value later.

Player 1 with Queen (middle card):

• First action: Always Check (betting dominated)
• After Check-Bet: Mix of Fold and Call

Queen plays depend heavily on opponent tendencies.

Why Mixed Strategies?

Pure strategies are exploitable. If Player 1 never bluffs with Jack:

• Player 2 can always fold to bets, never losing extra chips to bluffs

If Player 1 always bluffs with Jack:

• Player 2 can always call bets, catching every bluff

The equilibrium mix makes Player 2 indifferent to calling/folding against P1’s bets, preventing exploitation.

Why Kuhn Poker Matters

1. Tractable Yet Non-Trivial

With only 12 information sets, the game is small enough for:

• Complete analytical solution
• Rapid algorithm testing
• Educational understanding

Yet rich enough to exhibit:

• Bluffing as emergent optimal behavior
• Mixed strategy equilibria
• Positional advantage

2. Algorithm Benchmarking

Kuhn Poker is the canonical benchmark for testing:

• CFR convergence rates
• MCCFR variant performance
• New equilibrium-finding algorithms

Performance comparison (iterations to ε-equilibrium):

3. Extensibility

Variants allow controlled complexity increases:

• Leduc Poker: 6-card deck, 2 betting rounds
• 3-player Kuhn: Multiplayer dynamics
• Larger decks: More cards increase state space

4. Theoretical Foundation

Results on Kuhn Poker often generalize to larger games, making it ideal for:

• Proving algorithmic properties
• Testing convergence bounds
• Validating implementation correctness

CFR Results

Convergence Example

Running vanilla CFR for 100,000 iterations yields:

• Game value: ≈ -0.0554 (actual: -1/18 ≈ -0.0556)
• Error: < 0.04%

Sample Equilibrium Strategy

After CFR training, Player 1’s strategy approximates:

Facing Check-Bet:

Exploitability Metrics

CFR variants on Kuhn Poker achieve very low exploitability:

Sources

1. Kuhn Poker - Wikipedia
2. CFR on Kuhn Poker - labml.ai
3. Vanilla CFR for Engineers
4. Kuhn Poker - Grokipedia
5. GitHub: Kuhn Poker CFR
6. Kuhn, H.W. (1950) - “A Simplified Two-Person Poker”