Poker and Game Theory

In order to play your hand in poker, you have to follow a strategy (even if the strategy is unconscious).  For any strategy you use, there is a correct counter-strategy.  Winning players figure out what strategy their opponents are using and adopt the right counter-strategy.  This is one of the reasons that poker is a game of skill.

Here is an illustration.  (Experienced poker players should ignore the oversimplifications I am making.) In No Limit Texas Hold’em (the version of poker you are mostly likely to see on TV or be able to play in a casino), you are dealt two cards face down, your “hole cards.” There is only a small percentage of hands that are good enough for you to make the first bet with (“open”).  Your “opening range” includes pairs — like 22, 77, KK (king-king), and AA (ace-ace) — but it also includes high, unpaired cards — like AK (ace-king), KQ (king-queen), AQ.

Suppose you are dealt AK as your “hole cards.”  You bet and everyone folds except for one “caller” (a person who matches your bet to stay in contention to win the hand).

Now the “flop” is dealt (three cards put face up on the table, which anyone can combine with their own cards to make the best possible five-card hand).  Suppose the flop comes A72.  You “hit” this flop, because the ace in your hand and the ace on the flop combine to give you a pair of aces, which is probably now the best hand.  You should now “bet for value.”  In other words, you are betting in the hopes of getting called by a hand weaker than your own.  (Perhaps your opponent called you with A5, and now has a pair of aces, but with a weaker “kicker” than yours.  In other words, your five-card poker hand is AAK72, while his is AA572.  You both have a pair of aces, but your king-7-2 beats his 7-5-2.)

However, approximately 2/3 of the time, you will not make a pair with either your ace or your king.  So suppose the flop comes T72 (ten-seven-deuce), “rainbow” (of three different suits).  You “missed” this flop with your AK (i.e., your cards do not combine with the flop to make a pair or better).  What should you do?

You should employ one of the most basic game theoretic strategies in hold’em:  the “continuation bet.”  Bet, even though you missed the flop.  Why?  A continuation bet is a kind of “bluff,” a bet designed to get a better hand to fold. Your opponent doesn’t know that you missed the flop, and most of the time your opponent missed the flop too.  So if you bet, he will probably fold, because he assumes that you have a stronger hand.  For example, suppose your opponent called your pre-flop bet with 66.  He actually has the best hand now.  His T7662 has a pair while your AKT72 is only “high card.”  But it will be very hard for him to call your bet, because for all he knows, you could have started out with a higher pair than his (perhaps you have pocket jacks as your hole cards), or you could have raised with AT and hit the ten.  You have executed a strategy that beats your opponent the vast majority of the time.  When you hit the flop, you probably have the best hand, so your value bet wins you money when it is called, and doesn’t lose you anything when it is not.  When you don’t hit the flop, your continuation bet bluff wins the pot the majority of times, because the majority of times your opponent didn’t hit the flop either.

Suppose you keep continuation-betting. After a while, a smart and observant opponent will catch on to what you are doing.  How?  Statistically, there just aren’t enough good flops (flops favorable to your hand) for you to be hitting them as often as you are betting them.  Consequently, a good player will eventually “float” you.  To float is to call a bet on the flop, with the intention of betting on the turn to win the hand.  So eventually you find that your continuation bets are no longer working.  You bet pre-flop, get called, bet the flop, but then get called again.  When you don’t have anything (other than high card) you check the turn, your opponent bets, and you lose the hand.  Your opponent has developed an effective counter-strategy to your continuation-bet strategy.

But now suppose that YOU are smart and observant.  You realize that your opponent cannot have hit the flop as often as he is calling your flop bets:  he must be floating you, at least some of the time.  Consequently, you start “double-barreling”:  you continuation bet the flop as a bluff, and then bluff again on the turn.  Your opponent has observed that you often bluff on the flop, but has not seen you bet frequently on the turn, so he assumes that when you bet the turn you actually have a strong hand. Your opponent folds to your turn bets.  You have executed a successful counter-strategy to your opponent’s floating.

Of course, your observant opponent will eventually notice that you are now betting the flop AND the turn more often than you could reasonable have a strong hand.  He will start to call your turn bets with weak hands, to counter your double-barreling strategy.  You will eventually notice and start to “triple-barrel.”  Your opponent will then adopt the counter-strategy of “calling light” on the river, meaning he will call with fairly weak hands, on the assumption that you are bluffing for three streets now.

What should you do?  Stop triple-barreling.  But when you have a strong hand, you can bet the flop, the turn, and the river.  Your opponent will call with weak hands, because you have trained him that you normally have an even weaker hand when you call. As a result, you will make a huge profit.

Of course, your observant opponent will realize that you have stopped bluffing and are now only value-betting.  He will stop floating you.

And this is the occasion for you to start continuation-betting again.

About The Doc

"The Doc" is a professor at Vassar College (USA). However, the views expressed in his blog and comments are not necessarily those of Vassar, its administration, or other employees, none of whom bears any responsibility for his opinions.
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