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Introduction

In this article

• Is the push-or-fold mode according to Nash ideal?
  
• Can a push or a call according to Nash be -EV?
• When and how you should adapt your game
  
• Do these adaptations depend on the size of the ICM effect (bubble factor)?
  

Good heads-up play is vital if you want to play SNGs successfully. As the blinds are usually very high by the time you reach the HU, you will mainly find yourself in situations in which you fold your hands preflop or push them all in - this is known as push-or-fold play.

So-called nash ranges (NRs) play an important role in this context. Since you will often be up against a fish in the heads-up, who has built his stack through crazy moves, you need to be cautious when using such nash ranges for your game.

This articles serves as a complement and an extension of the articles SAGE – Sit and Go Endgame System and Push or Fold in Heads-up.

It is not necessary to have read them in order to understand this article. The so-called SAGE is basically based on Nash equilibria; SAGE simplifies them in a way that they are somewhat easier to memorise. However, this also makes the system rather stiff.

Basics

Abbreviations
BB	Big blind	NCR	Nash calling range
BL	Blind level	NPR	Nash pushing range
CR	Calling range	NR	Nash ranges
ES	Effective stack	P/F	Push-or-Fold
EV	Expected value	PP	Pocket pair
HEV	Hand EV	PR	Pushing range
HU	Heads-up	REV	Range EV
ICM	Independent Chip Model	SB	Small blind

First of all we need to clarify some basic terminology:

The effective stack (ES) is an important parameter in the analysis of push-or-fold situations in heads-up. It refers to the size of the smaller of the two stacks before the blinds (and antes) are posted, measured in BBs. Therefore, the ES is always the same for both players.

The blind level (BL) is 500/1000; after you pay the SB, you are left with 5500 chips, your opponent in the BB has 8000 chips left. In this situation, ES = 6.

A range in poker refers to a certain selection of hands. When it comes to push-or-fold situations, particularly the "x % ranges" are relevant: They consist of the strongest x % of the 1326 possible starting hands.

However, this definition is not clear-cut since the relative strength of a hand always depends on the specific situation, and this is also true for push-or-fold situations.

If ES = 9, the calling range (CR) of 42.7% comprises the hands 22+, Ax+, K2s+, K4o+, Q4s+, Q8o+, J7s+, J9o+, T8s+, T9o, 98s. The + means that all hands with a better kicker (or bigger PPs) are also included in the range. A hand such as K4 is therefore part of the range, while 78s is not.

Compare this to the 43.3% pushing range (PR) for ES = 16. It consists of the hands 22+, Ax+, K2s+, K7o+, Q5s+, Q9o+, J6s+, J9o+, T6s+, T9o, 96s+, 98o, 85s+, 75s+, 65s, 54s. You see that K4, K5 and K6 are not included here, while 54s is.

The example illustrates that a range of 43% for a call with ES = 9 does not equal the range for a push with ES = 16. That's why we will always list the relevant hands in the following illustrations. The %-figures and the precise ranges are taken from the ICM calculator at www.holdemresources.net.

It's easy to understand that the following applies as well:

If you add a random hand to a % range, you will get a new % range when that hand is the strongest hand outside of the range. Vice versa, if you remove a hand from a % range, you will get a new % range when that hand is the weakest hand within the range.

You have already encountered the Independent Chip Model in some previous articles:

Independent Chip Model (ICM) , Examples and explanations for ICM and Advanced ICM.

A HU is characterised by the fact that ICM effects are no longer in place. That's why the HU in an SNG (as well as the HU in an MTT) is basically the same as the HU in a cash game, in which every chip is worth

[($ 1st place) - ($ 2nd place)] / (Total number of chips).

Whether you should push your hand or call a push in a HU therefore mainly depends on the following parameters:

• Your ES
• Your opponent's calling or pushing range

Aside from the HU, there is always a certain level of the ICM effect in MTTSs and SNGs. In SNGs, this effect tends to have its peak on the bubble and is therefore also known and measured as the bubble factor (see The Bubble Factor in different SnG formats).

Nash in the HU

Most of the theory and the examples in this section are based on the heads-up game of a 9-10 player SNG with ES <= 20 and without antes. However, the very same conclusions can generally be applied to the HU in an MTT or blind battles in a cash game. As usual, we will call your opponent Villain. All results and examples in this section are calculated with a (future game) edge of 0%. In many cases we will round the %-figures to natural numbers and 42.7% therefore becomes 43% for example.

The expected value (short: EV) refers to the average result in a randomised experiment. For the push-or-fold game in a heads-up situation, the EV regarding the parameter "Chips before the action minus average number of chips after the action (in BBs)" is relevant.

We will now distinguish between the range EV (REV) of a push (or call) and the hand EV of a push (or call) with a specific hand. Both depend on the blinds and your opponent's calling range (or pushing range).

You are in the BB with ES = 9, your opponent's calling range is 42.7% (22+, Ax+, K2s+, K4o+, Q4s+, Q8o+, J7s+, J9o+, T8s+, T9o, 98s):

• Your push with T8 would have an HEV of around 0.
• Your push with A2 would have an HEV of around 1 BB (see calculation in appendix).
• Your push with a range of 58.7% (22+, Kx+, Q2s+, Q5o+, J2s+, J8o+, T5s+, T8o+, 95s+, 97o+, 85s+, 87o, 74s+, 64s+, 53s+) would have an REV of around 0.5 BB (find more on this in the next section).

In general, the REV is calculated as follows:

(1) REV(Range) = Sum_{Hand from range} HEV(Hand)* P(Hand)

P(Hand) refers to the probability that you are dealt this hand.

In the above example, A2 is therefore an above-average hand in the pushing range because its HEV is higher than its REV.

A general conclusion is that the REV increases if you add a hand with +HEV to the range and decreases if you add a hand with -HEV to the range. A hand with 0HEV does not affect the REV.

If you plan to expand your range, it would make most sense to pick the strongest hand outside of the range first. This hand has the biggest HEV and therefore increases the REV more than any other. If even the strongest hand outside of the range is -HEV, adding this hand to the range will at least decrease the REV less than any other hand. If you opt to reduce the range, you should accordingly remove the weakest hands from the range first. As indicated in the "Ranges" section, this is how new % ranges emerge from other % ranges.

We will now look at how nash equilibria are defined. However, our definition will be tailored to the push-or-fold situation in a heads-up.

Imagine you are in a HU with an ES of, for example, 9BB. In this scenario you push with a range of x% and your opponent calls a push with y%. Therefore, both of you have a certain REV, which may very well differ.

A Nash equilibrium (also called nash ranges, NR) is made up of ranges (x%, y%) so that neither you nor your opponent can increase his REV by modifying his range while the other keeps his the same. x is then called Nash pushing range (NPR) and y Nash calling range (NCR) and you say that the player pushes (or calls) according to Nash.

Considering the theory from the previous section, we conclude:

NRs contain all those ranges that comprise all +HEV hands and no -HEV hands. Of course it is a requirement that the opponent calls according to Nash (or that he pushes according to Nash when it comes to NCRs).

What follows from this is that when your opponent deviates from the NR, your REV increases. This is because 'deviating' means that he adds a -HEV hand or removes a +HEV hand. As this lowers his REV, your REV rises.

Read more on REVs in the "Range EV of Nash ranges" section.

Finally it should be added that the NRs in the ICM model are basically only dependent on the ES. It doesn't play a role here if you play against a 10 BB stack or a 100 BB stack with 5 BBs.

It's quite simple to calculate the NRs illustrated in the previous section. All of the ranges were calculated with the free ICM calculater at www.holdemresources.net, but could also be calculated with PokerStrategy.com's ICM-Trainer for example. Small variations of 0.5% can occur between the different programmes, sometimes the results might differ by one hand. However, these slight difference don't play a significant role.

The following tables give an overview. For the version that includes antes, the antes are 10% of the BB. Other structures, such as that at PokerStars (antes of between 12.5% and 5% of the BB, depending on the level) lead to a slightly different result. If they are 5% of the BB, for example, you could take the average from both tables.

Table 1 (without ante):

ES	NPR	NCR
1	100.0%: Any two	100.0%: Any two
2	94.6%: 22+, 8x+, 72s+, 73o+, 62s+, 63o+, 52s+, 53o+, 42s+, 32s	96.4%: 22+, 8x+, 72s+, 73o+, 62s+, 63o+, 5x+, 42s+, 43o, 32s
3	84.6%: 22+, Tx+, 92s+, 93o+, 82s+, 84o+, 73s+, 75o+, 63s+, 65o, 53s+	96.4%: 22+, 8x+, 72s+, 73o+, 62s+, 63o+, 5x+, 42s+, 43o, 32s
4	78.3%: 22+, Jx+, T2s+, T3o+, 92s+, 95o+, 83s+, 85o+, 74s+, 76o, 64s+, 53s+	82.2%: 22+, Tx+, 92s+, 95o+, 82s+, 85o+, 73s+, 75o+, 63s+, 65o, 53s+, 43s
5	74.1%: 22+, Jx+, T2s+, T5o+, 93s+, 96o+, 84s+, 86o+, 74s+, 76o, 64s+, 53s+	68.6%: 22+, Qx+, J2s+, J3o+, T2s+, T6o+, 94s+, 96o+, 85s+, 87o, 75s+, 65s
6	70.1%: 22+, Qx+, J2s+, J4o+, T2s+, T6o+, 94s+, 97o+, 84s+, 86o+, 74s+, 76o, 64s+, 53s+	59.6%: 22+, Qx+, J2s+, J6o+, T5s+, T7o+, 96s+, 98o, 86s+
7	66.2%: 22+, Qx+, J2s+, J6o+, T3s+, T7o+, 95s+, 97o+, 84s+, 87o, 74s+, 76o, 64s+, 53s+, 43s	50.8%: 22+, Kx+, Q2s+, Q5o+, J5s+, J8o+, T7s+, T8o+, 97s+
8	62.6%: 22+, Kx+, Q2s+, Q3o+, J2s+, J7o+, T4s+, T7o+, 95s+, 97o+, 85s+, 87o, 74s+, 64s+, 53s+	48.4%: 22+, Kx+, Q2s+, Q6o+, J6s+, J8o+, T7s+, T9o, 98s
9	58.7%: 22+, Kx+, Q2s+, Q5o+, J2s+, J8o+, T5s+, T8o+, 95s+, 97o+, 85s+, 87o, 74s+, 64s+, 53s+	42.7%: 22+, Ax+, K2s+, K4o+, Q4s+, Q8o+, J7s+, J9o+, T8s+, T9o, 98s
10	56.6%: 22+, Kx+, Q2s+, Q7o+, J3s+, J8o+, T5s+, T8o+, 95s+, 97o+, 85s+, 87o, 74s+, 64s+, 53s+	38.5%: 22+, Ax+, K2s+, K6o+, Q6s+, Q8o+, J8s+, J9o+, T9s
11	54.4%: 22+, Kx+, Q2s+, Q8o+, J4s+, J8o+, T5s+, T8o+, 95s+, 98o, 85s+, 87o, 74s+, 64s+, 53s+	36.3%: 22+, Ax+, K2s+, K6o+, Q7s+, Q9o+, J8s+, JTo, T9s,
12	51.4%: 22+, Ax+, K2s+, K3o+, Q2s+, Q8o+, J5s+, J9o+, T6s+, T8o+, 95s+, 98o, 85s+, 87o, 75s+, 64s+, 54s	33.0%: 22+, Ax+, K4s+, K8o+, Q8s+, Q9o+, J9s+, JTo
13	49.0%: 22+, Ax+, K2s+, K4o+, Q3s+, Q8o+, J5s+, J9o+, T6s+, T8o+, 96s+, 98o, 85s+, 75s+, 64s+, 54s	31.8%: 22+, Ax+, K5s+, K8o+, Q8s+, QTo+, J9s+, JTo,
14	49.0%: 22+, Ax+, K2s+, K5o+, Q4s+, Q9o+, J5s+, J8o+, T6s+, T8o+, 95s+, 98o, 85s+, 87o, 75s+, 64s+, 54s	30.0%: 22+, Ax+, K6s+, K8o+, Q9s+, QTo+, JTs
15	45.7%: 22+, Ax+, K2s+, K6o+, Q4s+, Q9o+, J6s+, J9o+, T6s+, T8o+, 96s+, 98o, 85s+, 75s+, 64s+, 54s	28.4%: 33+, Ax+, K7s+, K9o+, Q9s+, QTo+, JTs
16	43.3%: 22+, Ax+, K2s+, K7o+, Q5s+, Q9o+, J6s+, J9o+, T6s+, T9o, 96s+, 98,o 85s+, 75s+, 65s, 54s	27.1%: 33+, A2s+, A3o+, K8s+, K9o+, Q9s+, QTo+, JTs
17	43.3%: 22+, Ax+, K2s+, K8o+, Q5s+, Q9o+, J6s+, J9o+, T6s+, T8o+, 96s+, 98o, 85s+, 75s+, 65s, 54s	25.0%: 33+, A2s+, A4o+, K8s+, K9o+, QTs+, QJo, JTs
18	42.1%: 22+, Ax+, K2s+, K8o+, Q5s+, Q9o+, J7s+, J9o+, T6s+, T9o, 96s+, 98o, 85s+, 75s+, 65s, 54s	23.8%: 33+, A2s+, A4o+, K9s+, KTo+, QTs+, QJo, JTs
19	40.9%: 22+, Ax+, K2s+, K9o+, Q5s+, Q9o+, J7s+, J9o+, T6s+, T9o, 96s+, 98o, 86s+, 75s+, 65s, 54s	22.6%: 33+, A2s+, A5o+, K9s+, KTo+, QTs+, QJo
20	40.9%: 22+, Ax+, K3s+, K9o+, Q5s+, Q9o+, J6s+, J9o+, T6s+, T9o, 96s+, 98o, 86s+, 75s+, 65s, 54s	21.7%: 33+, A2s+, A5o+, K9s+, KTo+, QTs+

Table 2 (with ante):

ES	NPR	NCR
1	100.0%: Any two	100.0%: Any two
2	98.2%: 22+, 5x+, 42s+, 43o, 32s	100.0%: Any two
3	84.6%: 22+, Tx+, 92s+, 93o+, 82s+, 84o+, 73s+, 75o+, 63s+, 65o, 53s+	100.0%: Any two
4	84.9%: 22+, Tx+, 92s+, 93o+, 82s+, 84o+, 73s+, 75o+, 63s+, 65o, 53s+, 43s	89.4%: 22+, 9x+, 82s+, 84o+, 72s+, 74o+, 62s+, 64o+, 52s+, 54o, 43s
5	78.6%: 22+, Jx+, T2s+, T3o+, 92s+, 95o+, 84s+, 86o+, 74s+, 76o, 63s+, 65o, 53s+, 43s	73.2%: 22+, Jx+, T2s+, T4o+, 92s+, 96o+, 84s+, 86o+, 75s+, 65s
6	71.9%: 22+, Qx+, J2s+, J3o+, T2s+, T6o+, 94s+, 96o+, 84s+, 86o+, 74s+, 76o, 64s+, 53s+	65.6%: 22+, Qx+, J2s+, J4o+, T3s+, T6o+, 95s+, 97o+, 85s+, 87o, 76s
7	71.9%: 22+, Qx+, J2s+, J4o+, T2s+, T6o+, 93s+, 96o+, 84s+, 86o+, 74s+, 76o, 63s+, 53s+, 43s	56.3%: 22+, Kx+, Q2s+, Q4o+, J2s+, J7o+, T6s+, T7o+, 96s+, 98o, 87s
8	66.2%: 22+, Qx+, J2s+, J6o+, T3s+, T7o+, 95s+, 97o+, 84s+, 87o, 74s+, 76o, 64s+, 53s+, 43s	50.8%: 22+, Kx+, Q2s+, Q5o+, J5s+, J8o+, T7s+, T8o+, 97s+
9	65.6%: 22+, Qx+, J2s+, J7o+, T3s+, T7o+, 95s+, 97o+, 84s+, 87o, 74s+, 76o, 63s+, 53s+, 43s	49.6%: 22+, Kx+, Q2s+, Q6o+, J5s+, J8o+, T7s+, T8o+, 98s
10	60.8%: 22+, Kx+, Q2s+, Q5o+, J2s+, J7o+, T4s+, T8o+, 95s+, 97o+, 85s+, 87o, 74s+, 76o, 64s+, 53s+	43.0%: 22+, Ax+, K2s+, K4o+, Q4s+, Q8o+, J7s+, J9o+, T7s+, T9o, 98s
11	57.5%: 22+, Kx+, Q2s+, Q6o+, J3s+, J8o+, T5s+, T8o+, 95s+, 97o+, 85s+, 87o, 74s+, 64s+, 53s+	39.7%: 22+, Ax+, K2s+, K5o+, Q6s+, Q8o+, J8s+, J9o+, T8s+
12	54.4%: 22+, Kx+, Q2s+, Q8o+, J4s+, J8o+, T5s+, T8o+, 95s+, 98o, 85s+, 87o, 74s+, 64s+, 53s+	36.3%: 22+, Ax+, K2s+, K6o+, Q7s+, Q9o+, J8s+, JTo, T9s
13	54.4%: 22+, Kx+, Q2s+, Q8o+, J4s+, J8o+, T5s+, T8o+, 95s+, 98o, 85s+, 87o, 74s+, 64s+, 53s+	34.5%: 22+, Ax+, K3s+, K7o+, Q8s+, Q9o+, J9s+, JTo, T9s
14	51.4%: 22+, Ax+, K2s+, K3o+, Q2s+, Q8o+, J5s+, J9o+, T6s+, T8o+, 95s+, 98o, 85s+, 87o, 75s+, 64s+, 54s	33.0%: 22+, Ax+, K4s+, K8o+, Q8s+, Q9o+, J9s+, JTo
15	49.0%: 22+, Ax+, K2s+, K5o+, Q3s+, Q9o+, J4s+, J9o+, T6s+, T8o+, 95s+, 98o, 85s+, 87o, 74s+, 64s+, 54s	30.6%: 22+, Ax+, K6s+, K8o+, Q8s+, QTo+, J9s+
16	49.0%: 22+, Ax+, K2s+, K5o+, Q4s+, Q9o+, J5s+, J8o+, T6s+, T8o+, 95s+, 98o, 85s+, 87o, 75s+, 64s+, 54s	30.0%: 22+, Ax+, K6s+, K8o+, Q9s+, QTo+, JTs
17	45.7%: 22+, Ax+, K2s+, K6o+, Q4s+, Q9o+, J6s+, J9o+, T6s+, T8o+, 96s+, 98o, 85s+, 75s+, 64s+, 54s	28.4%: 33+, Ax+, K7s+, K9o+, Q9s+, QTo+, JTs
18	44.2%: 22+, Ax+, K2s+, K7o+, Q5s+, Q9o+, J6s+, J9o+, T6s+, T8o+, 96s+, 98o, 85s+, 75s+, 65s, 54s	27.1%: 33+, A2s+, A3o+, K8s+, K9o+, Q9s+, QTo+, JTs
19	43.3%: 22+, Ax+, K2s+, K8o+, Q5s+, Q9o+, J6s+, J9o+, T6s+, T8o+, 96s+, 98o, 85s+, 75s+, 65s, 54s	25.0%: 33+, A2s+, A4o+, K8s+, K9o+, QTs+, QJo, JTs
20	44.5%: 22+, Ax+, K2s+, K8o+, Q5s+, Q9o+, J6s+, J9o+, T6s+, T8o+, 96s+, 98o, 85s+, 87o, 75s+, 64s+, 54s	23.8%: 33+, A2s+, A4o+, K9s+, KTo+, QTs+, QJo, JTs

Every ES has its own line. In column 2 you can find the corresponding NPR (in the SB) and in column 3 the corresponding NCR (in the BB).

Here you can find another table, which basically contains the same information, but arranged differently. The advantage of this table is that it's more concise. An advantage of the above table is that you get a better feel for the ranges of the individual ESs. For example, it's easier for you to see how many weaker hands you could still push or how much better your hand would have to be for you to call.

We will now take a closer look at the actual REC of NRs. As the exact value is not that important for an ideal game, you can skip this section if you like. The only important thing is to pick a range with the biggest REV possible.

First of all, playing a hand in Texas Hold'em after paying the blinds is no longer a classic zero-sum game. All in all, you can win the blinds plus the sum of the opponent's winnings and losses. It would only be a zero-sum game if you never had to pay any blinds. In this scenario, however, there would only be one reasonable way to play: Don't play at all or only push and call AA. Not a game with big edges.

In the sections "Hand EV and Range EV" and "Nash equilibria in a HU" you already saw that all hands in the NRs are +HEV (or 0HEV). What does it mean now that the hand T8 in example 3 has an HEV of almost 0? It means that you will win close to nothing if you push against your opponent's calling range of 43%.

Vice versa this means that the your opponent's calling range has an REV of almost 1.5 BB if you push with T8. As he folds in 57% of cases and doesn't win any chips, he has to win an average of 1.5 BB / 43% = 3.5 BBs in the remaining cases. You therefore lose an average of 3.5 - 1.5 BB = 2 BB if you get called.

If you have AA instead of T8, the situation looks better of course. Your push has an HEV of around 3.8 BB, composed of 1.5 BB in the 57% of cases in which you don't get called and 43% in which you do get called, with an EV of around 6.9 BB.

All other hands have an EV of between 0 and 3.8 BBs.

You could go ahead and calculate the REV as we did in formula (1). This is a bit tedious, however. Another option would be to find the result in PokerStrategy.com's ICM trainer: The equity before placing the blinds is always around the same as the equity after the hand (if both opponents are playing according to Nash). For ES <7.5 BBs, the SB's REV is bigger than zero. For bigger ESs, it is smaller than zero, somewhere around 0.1 BB for ES = 15 to 20. This means that the SB's REV is 0.5 BBs and the BB's REV is 1 BB. What's interesting is that when it comes to the EV in the push-or-fold play, it is pretty much irrelevant whether you are in the SB or in the BB in the next hand.

Application and modification of Nash ranges

What implications does it have for yout game if you strictly adhere to NRs? In the section "Ranges" you already saw that the REV of an NR grows if your opponent deviates from the NR. There is no way for him to get your REV to sink below the value he reaches if he also plays according to Nash. That is what is meant when people say "Playing according to Nash is not exploitable".

However, only few of your opponents will play strictly according to Nash when in a HU situation. If your opponent plays with a different range, though, this also means a different HEV for your hands. This in turn leads to the circumstance that not all hands within the NR still have to be +EV and that there can be hands outside of the NR that are +EV.

Let's look at some examples, in all of which ES = 9. Therefore, NPR = 59% and NCR = 43%. We will distinguish whether you are in the SB or in the BB and whether you have a strong hand from your NR, a weak hand from your NR or a hand that is not included in your NR. For each scenario we will look at the change in the HEV (in the BB) in correlation to your opponent's range. More information on the exact calculations can be found in the appendix.

You are in the SB with ES = 9.

• Case 1: Strong hand from the Nash pushing range:

Calling range	20%	30%	43%	55%	70%
HEV	0.93	0.96	1.09	1.19	1.36

• Case 2: Weak hand from the Nash pushing range:

Calling range	20%	30%	43%	55%	70%
HEV	0.76	0.48	0.08	-0.26	-0.64

• Case 3: Hand not included in the Nash pushing range:

Calling range	20%	30%	43%	55%	70%
HEV	0,68	0,3	-0,18	-0,54	-0,68

You are in the BB with ES = 9.

• Case 4: Strong hand from the Nash calling range:

Pushing range	25%	40%	59%	70%	80%
HEV	0.39	1.45	2.13	2.33	2.49

• Case 5: Weak hand from the Nash calling range:

Pushing range	25%	40%	59%	70%	80%
HEV	-1.66	-1.29	0.125	0.68	1.06

• Case 6: Hand not included in the Nash calling range:

Pushing range	25%	40%	59%	70%	80%
HEV	-1.54	-1.43	-0.56	0.09	0.56

The examples illustrate the following trends:

• If you have a strong hand and push, your HEV increases the looser Villain calls.
• If you have a weak hand and push, your HEV decreases the looser Villain calls.
• If you call yourself, the HEV grows the looser Villain pushes, no matter if you have a strong or a weak hand.

Therefore, if you always play according to Nash, you are making a mistake in the following cases:

• Mistake 1: Case 2, CR = 55% and CR = 70%: You are making a -HEV push.
• Mistake 2: Case 3, CR = 20% and CR = 30%: You are skipping a +HEV push.
• Mistake 3: Case 5, PR = 25% and PR = 40%: You are making a -HEV call.
• Mistake 4: Case 6, PR = 70% und PR = 80%: You are skipping a +HEV call.

In these cases you should modify your range.

As explained in the "Hand EV and Range EV" section, you want to push or call those hands that are +HEV. From the trends illustrated in the previous example, we can set up the following guidelines for modifying your ranges (in accordance with the mistakes 1-4):

• Rule 1:  If Villain calls too loosely according to Nash, you should not push some of the weakest hands in the NPR. With these weak hands, your push might be -HEV.
• Rule 2: If Villain calls too tightly according to Nash, you can add some weaker hands that aren't yet included in your NPR to the range. Otherwise you might miss out on some +HEV pushes.
• Rule 3: If Villain pushes too tightly according to Nash, you should not call some of the weakest hands in the Nash CR. With these weak hands, your call might be -HEV.
• Rule 4: If Villain pushes too loosely according to Nash, you can add some of the stronger hands that aren't yet included in your NCR to your range. Otheriwse you might miss out on some +HEV calls.

In general you are more likely to face opponents that are too tight (cases 2 and 4), but rarely regulars that strictly play according to Nash, and rarely opponents who are on tilt for example and that push and call even more loosely than according to Nash (cases 1 and 3).

That's why the following ranges seem suited for default play against unknown players: Push approximately according to Nash. Call a bit tighter than you would according to Nash until you are convinced that the opponent pushes at least according to Nash.

If you are pushing and calling a bit too tight against a player who is way too tight, it has the following advantage: You decrease the probability that there will be a flip. This increases the average length of the game and you have more time to profit from your opponent's leak.

You have 2,800 chips, your opponent has 10,700, the blind level is 250/500 and you are in the SB. In the next seven hands you make three successful pushes, fold once and every time the opponent is in the SB, he is giving you the blinds. You now have 6,550 chips, your opponent has 6,950 and the blind level increases to 300/600. Villain pushes with ES = 11. Instead of the NPR of 54% you only give him a considerably tighter 28% and ask yourself if you should call with A7. According to ICM Wizard the call would be slightly +EV. However, your opponent shows a huge leak in the form of ranges that are way too tight. You should therefore fold in order to keep profiting from this.

To sum up, the game according to Nash shows the following traits:

• You get a positive REV.
• The REV grows if your opponent deviates from the NRs.

But:

• Your REV is no longer ideal.
• Some hands might even be -HEV.

You should therefore adjust the ranges. If you only play one hand against Villain, you certainly don't want it to be -HEV, even though the REV was positive.

Situations outside of heads-up

Many of the previous results can be easily applied to situations outside of the HU, unless the ICM effect is present. This is the case in cash games or winner-takes-it-all MTTs or SNGs.

In general, you can also calculate NR constellations for 3-10 players with the ICM calculator or the ICM trainer. The above rules of modification will still apply in this new context.

BU = 3200, SB = BB = 5000, BL is 200/400. The first player gets a ticket, the remaining seats don't.

The ICM calculator gives the following NRs:

PU	CA	OC	Range
BU			35.4%: 22+, Ax+, K4s+, K9o+, Q8s+, QTo+, J7s+, JTo, T7s+, T9o, 97s+, 86s+, 76s, 65s
	SB		20.7%: 22+, A2s+, A7o+, K9s+, KTo+, QTs+, JTs
		BB	10%: 55+, A9s+, ATo+, KQs
	BB		29.1%: 22+, Ax+, K7s+, K9o+, Q9s+, QTo+, J9s+
SB			50.8%: 22+, Ax+, K2s+, K4o+, Q2s+, Q8o+, J5s+, J9o+, T6s+, T8o+, 95s+, 98o, 85s+, 87o, 74s+, 64s+, 54s
	BB		32.1%: 22+, Ax+, K4s+, K8o+, Q8s+, QTo+, J9s+, JTo

This mean you can also push K4s from the BU, for example, unless you put at least one of the opponents on a looser calling range without putting the other one on a tighter range. To be more precise: Without putting the other one on a range that is so much tighter that it overcompensates the looser range of the other player. In this case you should use rule 1, i.e. push tighter and skip the push with K4s.

More interesting are situations in which the ICM effect or the bubble factor (BF) is bigger than 0.5. These are situations in which the chips won are worth less than the chips lost. The following equation applies here:

BF= $value (stack) / $value (2*stack).

In HU situations the bubble factor therefore is 0.5, in extreme ICM situations it can approximate 1. Typical examples of situations with a high bubble factor are the final table in an MTT, a qualifier right before reaching the ticket or the bubble of an SNG.

We are now taking a closer look at the bubble of an SNG. The conclusions drawn can then be applied to the other two situations. For example 8, we are adding a player in the CO with a small stack to example 7.

CO = 1800, BU = 3200, SB = BB = 5000, BL ist 200/400. CO foldet.

We get the following NRs:

PU	CA	OC	Range
BU			25.2%: 22+, A2s+, A8o+, A5o, K8s+, KTo+, Q8s+, QTo+, J8s+, JTo, T9s
	SB		7.1%: 88+, ATs+, AJo+
		BB	1.4%: QQ+
	BB		8.9%: 66+, ATs+, ATo+
SB			100%: Any two
	BB		3.2%: 88+

The first thing you see is that the NRs have become much tighter. This doesn't always have to be like this. If, for example, you play as a big stack, it is very well possible that your NPRs are bigger than in the comparable situation without the ICM effect. What happens if you are in the SB and wonder if you should push, the BB is a fish who has never heard about ICM and calls much too loosely, let's say with 10% of his hands?

Initially your push with weak hands from the range becomes -HEV again. But the ICM effect might now lead to the following:

The REV of your NR might decrease if the opponent deviates from his NR.

This is because the other two players profit if the BB calls you too loosely. This means the BB has a considerably lower REV with his CR of 10% than with the NCR, but his REV might also be reduced.

If you don't adjust the NRs, your REV could even become negative.

The modification rules from section 4.2 therefore have to be modified themselves, according to the following rules:

• Rule 1: The more the range is restricted, the higher the BF.
• Rule 2: The less the range is extended, the higher the BF.
• Rule 3: The more the range is restricted, the higher the BF.
• Rule 4: The less the range is extended, the higher the BF.

Appendix

In the examples 3-5 the HEV is given in BBs. You can use the programme SNG Wizard to calculateit: At first you choose the stacks and the blinds in a way that you get the desired ES. With an ES of 9, for example, you choose blinds of 300/600, SB=5400 and BB=9600. Then you select a specific hand for the SB, a CR for the BB and get the equity difference. To convert this to the unit BB, you have to multiply this with

All chips in the hand (in BBs) / 20

In the above case this would be a factor of 25/20 =1.25.

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