Kuhn Poker : How a 3-Card Game Reveals the Art of Bluffing
A brief overview of this unique 3-card poker variant
An Introduction to Kuhn Poker
As a game theorist who loves both math and poker, I’m excited to introduce Medium readers to the fascinating game of Kuhn poker. While it may sound complex at first, my goal is to explain the basics in simple, easy-to-grasp language that makes this unique game accessible to anyone.
As someone who cut my teeth on classic games like Go in the streets of South Korea, I was fascinated when I recently discovered Kuhn Poker. Developed by mathematician Harold W. Kuhn in 1950, this stripped-down version of poker has only 3 cards yet still manages to demonstrate important strategic concepts that apply to more complex games.
The Deceptively Simple Setup
Kuhn Poker is played with just three cards in the deck — a King, Queen and a Jack. Yup, this has to be the most bare bones card game you ever encountered.
Two players ante $1 each to play, and are both dealt one card randomly from the deck. This leaves only card in the deck, that is placed aside.
With such a small deck and short gameplay, you’d think successful strategies would be straightforward. You would be far from the truth !
Betting Rounds Distilled to a Pure Form
The first player can open the betting round by checking or betting $1. Easy so far.
If they bet, their opponent can call, match the $1, or fold. If the first player checks, the opponent can still open betting for $1 or also check.
When a player folds, the other wins the pot, just like in Texas Hold’em.
If neither folds by the end, whoever has the higher card wins.
This is poker 101.
Mixing Math and Bluffing
Yet despite seeming simple on the outside, Kuhn Poker is actually game of mixed-strategy Nash equilibrium. Wait, what ?
This means using chance, not just skill, is part of optimal play. Simply put, there is no pure strategy equilibrium. There is no optimal card counting strategy like in blackjack.
For example, say I have the Jack. Math shows I should bet a minority of the time and check a majority of the time to mimize my odds of losing equity long term.
But I can also bluff! If my opponent folds to my random bets, I win even with the lower Jack. Just like in poker right ?
So the beauty of Kuhn Poker is distilling gaming to its essence — math, psychology and skill all intersect to create a worthy challenge.
Randomization using mixed strategies (example)
Ok this may sound complex, but hear me out for a bit.
In game theory, randomization using mixed strategies is key to gain an edge over your opponents. This is assuming you are playing a game of imperfect information like poker.
Pure strategies create weaknesses (because they are exploitable), while mixed approaches lead to an optimal probabilistic balance. With the latter, players cannot expect to do better by unilaterally deviating. Similar to GTO (Game theory optimal) in poker. You become unexploitable.
For example in Kuhn poker, the player with the Queen can employ a mixed strategy Nash equilibrium, randomly folding a certain percentage of the time and calling the rest of the time. Only the player knows this percentage, AND, the player can adjust this ratio at random times.
For example if he sees the opponent doing a certain action, like looking at this watch, he can add 5% every time. It he sees the color red, he can change the ratio. If he sees the player twitch, etc.
This balanced strategy prevents the opponent from exploiting predictable patterns. And it is impossible to decipher.
In Closing
While simple in form, Kuhn poker allows for creative bluffing, calculated risks, and leveraging position and one’s environment— proving there is skill beyond just the luck of the draw.
Let me know in the comments if you have any other questions. I’m happy to explain further until Kuhn poker clicks for you. And who knows, you might discover a new favorite game!
