## what you need to know about equilibrium with …

Nash equilibrium is one of the main concepts in game theory, according to which a player cannot increase his winnings by changing strategies if his opponents do not change their strategies.

There is a Nash equilibrium at any stage of the game. However, changing the strategy based on it has the greatest effect at the push / fold stage. This is due to the fact that at other stages the factors influencing each distribution are much larger and it is more difficult to use the balance. Because of this, in poker, when people talk about Nash equilibrium play, they usually mean push / fold – we will also cover it in this article.

## Nash Push Fold: Equilibrium Strategy

Although theoretically Nash push / fold is applicable both in tournaments and in classic cash, in practice in the latter it is almost impossible to use it due to the depth of the stacks – they rarely fall to 15 BB and below. However, in some cash formats you still need to rely on Nash – for example, All-in or Fold or at tables with limited stack size.

To get the most value out of a 15bb or lower stack, you need to know two things:

- What ranges are optimal for a particular situation (they are called Nash ranges or equilibrium ranges);
- How close is the real range of each opponent to optimal and in which direction he deviates (tight or loose).

Basic Nash ranges are calculated under ideal conditions – assuming that all opponents at the table are familiar with Nash equilibrium and are considering it. In reality, such situations almost never happen, since people are either not familiar with this concept at all, or do not fully adhere to it.

If you rely only on basic ranges and do not take into account the real play of your opponents, then at a distance you will not profit in push / fold situations, because you will play sub-optimal.

However, you need to know the Nash ranges as they are the starting point when adjusting your strategy to suit your opponent’s style.

Consider an example: in the small blind your stack is 13.3 big blinds, in the game it’s just you and the big blind. According to Nash, in such a situation, we:

- Push – all pocket pairs, AX, KXs, Q3s +, J4s +, T6s +, 94s +, 84s +, 74s +, 64s +, 54s, A2o +, K5o +, Q9o +, J8o +, T8o +, 98o, 87o.
- Call – all pocket pairs, AX, K6s +, Q9s +, J9s +, A2o +, K8o +, Q9o +.

However, if you know that your opponent in the big blind has a tighter range – he deviates from the Nash equilibrium – then you also need to deviate with this information in order to extract more value. So your range of hands will need to be widened to push and narrowed to call. This is the only way you can exploit this opponent’s weakness.

Remember that push / fold ranges need to be adjusted depending on your position and how close you are to the prizes and the final table (the latter – except in situations where there is only one prize in the tournament). The change when approaching the prizes is due to the fact that among the player’s tasks there is not only stack accumulation, but also survival – then the basic equilibrium strategy is complemented by ICM.

To appreciate the importance of taking position into account when determining a push / fold range, we performed ChipEv calculations for you using the ICMIZER preflop calculator. In the screenshot below, you can see how the calling range of a 6-max Sit & Go changes with an effective depth of 8 BB, depending on whether you are sitting on the button or in the big blind.

You can also check out Nash’s basic ranges below – we’ve taken the small blind versus the big blind heads-up with no antes as examples. Suited hands are in bold.

### Nash table: pushing SB vs BB depending on the stack in the big blind

A | K | Q | J | T | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | |

A | 20+ | 20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |

K | 20+ | 20+ | 20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
19.9 |
19.3 |

Q | 20+ | 20+ | 20+ | 20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
16.3 |
13.5 |
12.7 |

J | 20+ | 20+ | 20+ | 20+ | 20+ |
20+ |
20+ |
20+ |
18.6 |
14.7 |
13.5 |
10.6 |
8.5 |

T | 20+ | 20+ | 20+ | 20+ | 20+ | 20+ |
20+ |
20+ |
20+ |
11.9 |
10.5 |
7.7 |
6.5 |

9 | 20+ | 20+ | 20+ | 20+ | 20+ | 20+ | 20+ |
20+ |
20+ |
14.4 |
6.9 |
4.9 |
3.7 |

8 | 20+ | 18 | 13 | 13.3 | 17.5 | 20+ | 20+ | 20+ |
20+ |
18.8 |
10.1 |
2.7 |
2.5 |

7 | 20+ | 16.1 | 10.3 | 8.5 | 9 | 10.8 | 14.7 | 20+ | 20+ |
20+ |
13.9 |
2.5 |
2.1 |

6 | 20+ | 15.1 | 9.6 | 6.5 | 5.7 | 5.2 | 7 | 10.7 | 20+ | 20+ |
16.3 | * |
2 |

5 | 20+ | 14.2 | 8.9 | 6 | 4.1 | 3.5 | 3 | 2.6 | 2.4 | 20+ | 20+ |
** |
2 |

4 | 20+ | 13.1 | 7.9 | 5.4 | 3.8 | 2.7 | 2.3 | 2.1 | 2 | 2.1 | 20+ | *** |
1.8 |

3 | 20+ | 12.2 | 7.5 | 5 | 3.4 | 2.5 | 1.9 | 1.8 | 1.7 | 1.8 | 1.6 | 20+ | 1.7 |

2 | 20+ | 11.6 | 7 | 4.6 | 2.9 | 2.2 | 1.8 | 1.6 | 1.5 | 1.4 | 1.4 | 1.4 | 20+ |

* 63s – for stacks 7.1 – 5.1 and 2.3

** 53s – for stacks 12.9 – 3.8 and 2.4

*** 43s – for stacks 10 – 4.9 and 2.2

### Nash table: SB vs BB call depending on the BB stack

A | K | Q | J | T | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | |

A | 20+ | 20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |
20+ |

K | 20+ | 20+ | 20+ |
20+ |
20+ |
20+ |
17.6 |
15.2 |
14.3 |
13.2 |
12.1 |
11.4 |
10.7 |

Q | 20+ | 20+ | 20+ | 20+ |
20+ |
16.1 |
13 |
10.5 |
9.9 |
8.9 |
8.4 |
7.8 |
7.2 |

J | 20+ | 20+ | 19.5 | 20+ | 18 |
13.4 |
10.6 |
8.8 |
7 |
6.9 |
6.1 |
5.8 |
5.6 |

T | 20+ | 20+ | 15.3 | 12.7 | 20+ | 11.5 |
9.3 |
7.4 |
6.3 |
5.2 |
5.2 |
4.8 |
4.5 |

9 | 20+ | 17.1 | 11.7 | 9.5 | 8.4 | 20+ | 8.2 |
7 |
5.8 |
5 |
4.3 |
4.1 |
3.9 |

8 | 20+ | 13.8 | 9.7 | 7.6 | 6.6 | 6 | 20+ | 6.5 |
5.6 |
4.8 |
4.1 |
3.6 |
3.5 |

7 | 20+ | 12.4 | 8 | 6.4 | 5.5 | 5 | 4.7 | 20+ | 5.4 |
4.8 |
4.1 |
3.6 |
3.3 |

6 | 20+ | 11 | 7.3 | 5.4 | 4.6 | 4.2 | 4.1 | 4 | 20+ | 4.9 |
4.3 |
3.8 |
3.3 |

5 | 20+ | 10.2 | 6.8 | 5.1 | 4 | 3.7 | 3.6 | 3.6 | 3.7 | 20+ | 4.6 |
4 |
3.6 |

4 | 20+ | 9.1 | 6.2 | 4.7 | 3.8 | 3.3 | 3.2 | 3.2 | 3.3 | 3.5 | 20+ | 3.8 |
3.4 |

3 | 20+ | 8.7 | 5.9 | 4.5 | 3.6 | 3.1 | 2.9 | 2.9 | 2.9 | 3.1 | 3 | 20+ | 3.3 |

2 | 20+ | 8.1 | 5.6 | 4.2 | 3.5 | 3 | 2.8 | 2.6 | 2.7 | 2.8 | 2.7 | 2.6 | 20+ |

## Nash Equilibrium and ICM

In tournaments, the players’ stacks are not the same as their equity, therefore, to calculate the mathematical expectation of getting into the prizes, an independent model of transferring the stack to equity – ICM – was invented. It is based on the Nash equilibrium, but it is much more complex than the basic concept due to the fact that it simultaneously takes into account four important factors:

- Prize structure;
- Opponents’ actions;
- The probability of each opponent being knocked out.

It is very difficult to calculate the odds, taking into account all possible situations and influencing factors on your own – to facilitate this task, poker calculators have been developed that work, among other things, on Nash’s mathematical calculations. However, unlike the concept of balance, calculators take into account very specific factors that further shift ranges – for example, the presence and size of knockouts.

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