Nash equilibrium

Nash equilibrium is a part of the games theory; it was written by American mathematician John Nash. This theory demonstrates the optimal game "in a vacuum": when to go all-in or to call the opponents' push. It is important to understand that Nash's push/call strategy in modern poker realities is no longer the only truthful one. It is optimal only if your opponents know about this strategy and stick to it without deflections.

The optimum use of the push/fold strategy by Nash can be done only against strong and understanding players. With a minimum deflection, the effectiveness of this strategy is being significantly reduced. The most profitable option of using Nash equilibrium is to adjust to the opponents and correct your own game on the basis of the opponents' ranges.

Where to use the Nash equilibrium?

The ranges of Nash equilibrium are suitable for playing in MTT, Sit & Go and Spin & Go tournaments. You should apply this strategy when your stack is being reduced to 15 big blinds or lower and your game is come down to only push/fold decisions. In order to master your game skills, you should use special software that simulates such situations: SNGWizard and ICMIZER.

Suppose your opponent goes all-in and you have 14 big blinds left. According to Nash equilibrium, you can call with a wide range of hands, having 20 BB, including pocket triples, QJ, QT and even K2s.

But it is a range "in a vacuum", which doesn't take into account the type of tournament, the stage and the difference in payouts. This strategy is mathematically correct but only if the game consists of two pre-flop decisions: push or fold. In modern realities, strong players can play a deep post-flop hand even with a stack of 15 big blinds.

Besides the use of Nash equilibrium, you can always just wait for a good hand and call the opponent. But if you don't know exactly what hand is good in reference to the size of your stack, Nash charts can help you to orient.

Usage of Nash equilibrium in the game will suit the novice players as it will provide an initial understanding of the push or call ranges for standard tournament situations and will help players to start earning money with poker fast enough.

Push range by Nash

According to Nash, it turns out that with 10 BB stacks we go all-in with the following hands:

• all pairs (even with 20 BB stacks);
• all suited aces, kings, queens, almost all suited jacks;
• all off-suited aces, kings;
• all suited connectors (according to Nash, it turns out to be profitable to push with stacks of 20 BB 54s);
• a number of other hands;

In reality, pushing depends primarily on your opponent and their hands for calling. 95% of opponents don’t call in accordance with Nash (all professional players adjust their ranges).

Call range by Nash

Nash Equilibrium is good for beginners. If your opponent pushes wider than Nash, you should call him wider. If your opponent is tight, you should also call him tight.

If your opponent makes wide calls (wider than according to Nash), you should push fewer hands than it is indicated in the table. If your opponent rarely calls all-ins, then you should go all-in more often than is indicated in the chart.

Remember, you don’t have to use Nash with stacks bigger than 15 BB. Nash Equilibrium doesn’t take into account that your opponent won’t shove with a number of hands. Your rival is unlikely to go all-in with 20 BB with top hands.

It is worth noting that Nash in its basic form is not acceptable for MTT. In tournaments, ICM has particular importance and almost all opponents play tighter than it is indicated in the tables.

Let's consider the example

There are 3 players in the game and only 2 of them are winning the chips. Blinds’ level is 120/ 240. The first player has 3 000 chips, we have 2 000 chips and the third participant has only one chip.

The player with one chip is on the button and he decides to fold. We are on SB.  Nash equilibrium will be relevant for us with an all-in probability of 13%, and 7.5% call probability from the first player on BB. Ranges are tight because you don't want to leave the game before the guy with one chip does. However, the player on BB is a fish and he is ready to call you with 7-6 suited or similar hands. Such a move is very disadvantageous for him, but we are also in a disadvantageous situation because all-in has too negative EV.

As for the victory, the player with one chip who has decided to fold becomes the winner. He gets a value despite one chip and fold.

This example shows that using Nash equilibrium ranges doesn’t guarantee you profit or even a break-even situation. Nash equilibrium only works if everyone plays without mistakes. If someone makes mistakes, it may have negative consequences for you.

Don't worry, because poker will always be the game in which you have to constantly adjust to your opponents’ play. There is no "secret mathematical strategy" that will allow you to always win.

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Nash equilibrium advantages

• If you play against the best players, they will use push/ fold range by Nash. That’s why you should be aware of Nash equilibrium as it is still very useful to know. By pressing one button, you can get the best strategy and professional poker players follow it
• The suggested push/ overcall ranges will work very well for almost any bet and structure. If you are a newbie in SNG, you can still make calculations based on the Nash equilibrium push/ overcall ranges and your game will improve immediately.
• Ranges are generated almost instantly.
• Nash equilibrium takes into account the stack sizes of all players and payout structure. It offers a good starting point for inexperienced players who still find it difficult to assign the ranges for their opponents in various situations.
• Nash calculator can be used for most tournament situations. This allows players to understand the optimal ranges for most of the possible spots.
• Nash equilibrium assigns optimal ranges not only for Hero, but also sets the overcall ranges for all the players. This saves much time and you don’t have to manually adjust your overcall ranges. Although all players will not follow Nash, the results will usually be close to the truth.

If you want to see how to apply Nash equilibrium in game, we offer you to watch the following video: