Three cards are enough


How many cards do you need to play a challenging game of poker? 52? 30? Actually, just three will do. Let’s take an Ace, a King and a Queen from a standard deck, and call our two players John and Tom. We’ll make them both put $1 into the pot, deal a card to each of them and tell John he has to check. Tom can then check if he wants to, and their two cards will be shown down. An Ace beats a King beats a Queen for the $2 in the pot.  Alternatively, Tom can bet $1, after which John can either fold, and let Tom have the $2 pot, or call the $1, and go to showdown. If you’ve not seen this game before, please think about what your strategy should be before you read on.

If Tom has the Ace he will bet, and if John has the Ace he will call a bet. The Ace is the nuts.

If John has the Queen, he will fold to a bet. He doesn’t beat anything.


Now we’re left with the more interesting decisions. First of all, what should Tom do with the King? If he bets, John will call with the Ace and fold the Queen. Tom can’t make John either fold a better hand or call with a worse hand, so he should check and show down his King. He may think he has the best hand, but betting is not the best play. This seems fairly straightforward, but it’s a concept that is poorly understood by many players. Real poker games are far more complex than three card poker, and it’s much harder to tell when you have the equivalent of a King – a hand with showdown value. The temptation to bet is often too strong, and if your opponent folds, well, you won the pot. The result gives you no feedback on your poor decision.

All that’s left to discuss is how Tom should play the Queen and what John should do with the King if Tom bets. If Tom has the Queen, he knows that he has the nut low. Should he bluff and try to get John to fold if he has the King? He’ll never make him fold the Ace. If John has the King and Tom bets, John knows he only beats a bluff, so should he call? If John calls whenever he has the King, Tom can exploit him by never bluffing with the Queen. If John folds whenever he has the King, Tom can exploit him by bluffing whenever he has the Queen.

If you show this problem to an expert in Game Theory, he’ll tell you that the solution of the game, by which he would mean a pair of strategies from which neither player can deviate without losing money, is for Tom to bluff one third of the time that he has the Queen and for John to call one third of the time that he has the King (I’ll spare you the maths). By using this strategy, Tom wins on average, and John therefore loses on average, about five and a half cents per hand. Tom has position on John, and can show down his Kings. John can’t realise the showdown value of his Kings and therefore loses in the long run.

That’s what a Game Theorist would say, but in reality, if this game is played repeatedly between two real people, things become more complex. John may deviate from this equilibrium strategy, perhaps bluffing with less than one third of his Queens in an attempt to lure John into folding more than one third of his Kings. If he can get John to fold too often, he can start to bluff more often…..until John starts to notice that Tom is bluffing too much, when he may call more often…and then Tom may start bluffing less often….and so on. A complex, dynamic interaction between the players can lead them to play ever-changing strategies in an attempt to exploit each other. This also assumes that both players are purely rational. What if John correctly calls with one third of his Kings, but Tom has a great run of cards, and repeatedly shows down the Ace? Maybe John will start calling more often because he feels that Tom ‘can’t always have it’, or start folding more often because Tom ‘is such a luckbox’. Real people can be irrational and stubborn, even before they start to tilt.

“ Many of the elements of the poker games that you play are present in this simple three card game – even some of the psychological elements. ”

Many of the elements of the poker games that you play are present in this simple three card game – even some of the psychological elements. If you add in all the complexities of a game like No Limit Hold’Em – for example, variable bet sizing, check-raising, four streets of betting and a board that changes on every street– you can start to see how complex a game poker is. However, if you can understand the Ace-King-Queen Game, you’ll have taken the first step on the road to understanding the theory of poker.


John Billingham

John Billingham

Professor John Billingham is a mathematician and keen recreational poker player with an interest in the mathematics of poker. He is the co-author, with Thomas Tiroch and Emanuel Cinca, of The Education of a Modern Poker Player, which was published last year by D&B Poker Books. The book documents his progress from fish to not quite as much of a fish under the expert guidance of his co-authors and provides the reader with plenty of sound, practical poker advice.
John Billingham

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