There are many theories and strategies that have surrounded the poker world. Some make sense and others, not so much. One that has been around for decades has been the Nash Equilibrium. This theory was created by mathematician, John Nash. Many poker players are familiar with the name John Nash; most think the Nash Equilibrium was a theory made just for poker. However, they could not be further from being right.
John Nash at a symposium of game theory at the university of Cologne in Germany
Born in 1928, Nash was a graduate of Princeton and MIT. In 1994 he would win the Noble Prize for Economic Science. His theories have been used to determine stock markets, economic outcomes and even used to estimate an outcome of a war. Outside of the scientific world, he may be best known for the movie, “A Beautiful Mind”. The film was based on his biography and the main character, played by Russell Crowe is, John Nash. Most of what Nash has contributed to the world is still being put to use today in some form.
I could go on and on about all that Nash created, as most of it is spectacular and intriguing. However, this is a poker site and you are reading this to learn about strategy, not how to determine the outcome of WWIII. It is important to understand why and how the Nash Equilibrium relates to poker. Let’s first discuss what this theory is all about.
The Nash Equilibrium is used to determine what decision will be best in certain situations. This theory states that the optimal outcome of a game is when no player benefits from changing their strategy taking into account their opponents decisions. For example, if two players both know the correct strategy of a game, then neither player will benefit by changing their decisions and Nash Equilibrium is present. Not every game will have a Nash Equilibrium present. Games such as monopoly and anything that involve chance will not apply to this theory.
A good example of this can be found in game shows. There was one in particular that pitted two players as a team at the beginning. For every question answered correctly, money would be added to a shared prize pool. At the end of the game both players would have to choose whether they would share the money, or take the money for themselves. If one picked share and the other picked take, the player who picked take got it all. If they both picked take, no one received a dime. And lastly, if they both picked share, they would split the money. Let’s assume player A and player B both tell each other they will pick share. This is the best possible outcome for both players and neither will benefit from changing their decision.
I know I stated that games of chance will not apply, and poker does have the element of luck involved. This theory will serve no purpose while playing a poker game with 3+ players should be ignored. However, it does specifically apply when heads-up in a sit-n-go (SNG)/tournament, more specifically when you are in push/fold mode.
In most turbo SNG’s and sometimes Multi Table Tournaments (MTTs), both players will be short stacked relative to the blinds. This means that a standard raise would pot commit the player making it difficult to play a normal style of poker. This is when push/fold comes into effect. It can be tricky to determine what hands to fold, shove or even call with in these situations. We can use the Nash theory to determine what an unexploitable play will be, given the circumstances.
If both players know the hole cards then each can make the best decision. For example if player A has 
and player B has 
, player B can make the best decision by folding and player A makes the best decision by going all-in. Neither player will benefit by changing their play. Obviously we aren’t going to be able to see our opponent’s hole cards and they won’t tell us what they will do. However, if we take into account their style of play we can use Nash theory to make the best decision for us. We can assume that a player will fold X hands and call with Y hands. Theoretically we will know what the best decision for us is to make.
So how do we use Nash to win at poker? Using Nash we formulate a chart of hands that will dictate what hands we shove, call and fold with. It will also determine at what blind levels to make each decision. This chart will show you exactly what hands are unexploitable to go all in at specific blind levels. Here is the best chart to use when playing. On the chart, the number in the cells refers to the number of big blinds you should have to push all-in with that hand.
There are some very important details that must be covered before you start using this theory. This concept is generally used when your opponent is better than you. If they make better decisions, then you can use Nash to make unexploitable plays that will decrease any advantage they may have. If a player is bad they will not follow the rules of this theory and shoving some of these hands will show a negative outcome. It can still be used as a guideline but should never be followed to a T.
Also, when I say that a play is “unexploitable”, this does not mean it is optimal. The term unexploitable indicates that the play you make cannot be changed by the decision of your opponent. If you shove Aces when the blinds are 15/30, this is unexploitable because no one can bluff you off the hand. It is far from optimal since you will likely get no callers and win little money.
By using these charts and having a solid foundation of SNGs you should be able to increase your win rates while playing SNGs (check out our SNG Strategy Guide for more info on playing Sit-N-Go’s). Heads-up is probably the most difficult for poker players since it is not played often enough. Changing just one 2nd into a 1st will have an enormous effect on your bankroll over the long run. For more detailed information on Nash and his theories you can read this pdf.
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Nice article on a very underestimated topic by most novice players. I think the key to Nash is to make players understand the difference between optimal and non exploitable play, you do a good job of this in this article Chris…
Jeff
Glad you enjoyed the article Jeff – completely agree that it’s a vital concept. The same goes for lots of the more “mathematical” side of poker unfortunately – few players spend the time to truly get to grips with it.