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The Nash Calculator and the Nash Equilibrium strategy in poker

by Mikey McDonald.

equilibrium in real life

Defining Nash Equilibrium

Nash Equilibrium in poker is when player’s ranges are in equilibrium with each other, or in other words, when opponents are playing an “optimal” game against one another whereby they both cannot gain anything by deviating from equilibrium strategy. This leads to an eventual stalemate (long term break-even proposition) and it makes no sense for either opponent to deviate from Nash if the other player is strictly following Nash – this means they will be deviating from “optimal” strategy and the other player may benefit. If one player deviates from Nash then the other player can do better by re-adjusting their ranges accordingly. Note that the word optimal remains inverted since following Nash will be far from optimal in a lot of situations as we’ll go on to later explore.

The Nash Equilibrium calculator

How it works

What the Nash calculator in ICMIZER does is take the pay-out, stack sizes and the blinds of the tournament into account to generate optimal ranges. This is useful since it allows you to understand what ranges are optimal in different spots and generally what players will be doing if they are adopting this strategy. It is also able to generate these optimal ranges for each individual player at the table and not just the hero or player in action. To carry out this calculation in the program all you have to do is hit the Calculate Nash Equilibrium button and the Nash ranges for each of the stacks will be assigned.

What makes this a good feature?

  • Instant results/simplicity – with the click of a button ranges are generated instantly and the feature is very simple to use.
  • If you would be playing with the very best players, they would be following Nash equilibrium push/fold strategy. While you most likely aren’t playing highest stakes 500$ hyper turbo structure, where most regulars are very aware of Nash strategy it still a good thing to know. Technically with a single button click you get presented the best possible strategy, which is actually used by pro poker players. That is just great! On the contrary if you would like to know the postflop strategy for top regulars of SNGs you would be out of luck, since that’s not something they like to share.
  • Suggested open Push and Overcall ranges, which you will get from Nash calculator, will be very good almost in any stakes or structures. If you are totally new to ICM you can just start using Nash equilibrium ranges for open pushing and overcalling, and your end game will improve immediately. Note that calling ranges against single open push suggested by Nash equilibrium strategy are likely to be pretty inefficient because your opponents aren’t likely to be pushing according to Nash.
  • Since Nash takes into account stack sizes relative to pay-out structure (essentially ICM) it offers a good baseline for new or inexperienced players who struggle to put opponents on ranges or don’t have a grasp for what an average range is in a spot. For example, the Nash range for pushing as the big stack into the mid-stack on the bubble will usually be very wide – this will help novice players understand they can push wide here. Similarly, a much tighter range will be recommended for shoving as the clear mid stack into the clear big stack on the bubble and so will help starting players learn some crucial ICM dynamics.
  • Flexibility of calculations – the Nash calculator can be used for the majority of the most common tournament/SNG situations that arise including open pushes, calling single pushes and calling multiple opponent pushes. This will help players understand optimal ranges for a wide array of spots.
  • Overcall ranges – Nash not only assigns optimal ranges for the player in action but incorporates overcall ranges for the other opponents and these ranges very often closely represent real population overcall ranges thus providing reasonably accurate results. This basically saves the user a lot of time from manually adjusting overcall ranges of players behind and whilst ranges won’t be 100% accurate as it’s unlikely each player will be following Nash, it does provide solid immediate results.

When to adopt Nash

One of the most common situations in poker where people adopt Nash ranges is HU at high blind levels against a tough opponent. This limits any edge your opponent has over you and any shove you make is unexploitable so can’t be wrong. If you revert to this style of play then your opponent will have to do the same to avoid being exploited. Usually, the caller will do better since their range is tighter so when stacks end up all-in the caller will have a stronger range; this however is balanced with the fact that the pusher picks up the blinds more often, so in the long run the number of chips lost and won will be equal therefore it’s a break-even proposition for both player’s. So why do it? Again, this is a strategy that should only be employed against very good players you perceive to be +EV vs you and will reduce any skill edge your opponent may have over you. Adopting the Nash range against an inferior opponent for instance who is letting you limp in a lot, steal the blinds pre-flop often and pick up pots on the cheap side post-flop will be far from optimal.

Limitations of Nash’s assumptions

Before electing to adopt Nash Equilibrium ranges, there are several important assumptions Nash makes that must be understood:

  • Nash strategy makes a very bold assumption that not only does your opponent know the Nash numbers (unlikely) but is also using them (unlikelier).
  • Nash strategy also makes another bold assumption that play is limited to push/fold which in the majority of cases is not true.

Elaborating on the first assumption, consider this example on the bubble of a 6max SNG:

Example 1

Blinds: 75/150

BTN: 500 folds

SB: 6000 shoves

BB: 2500

The Nash Equilibrium range for the SB is to push any two, and the Nash calling range for the BB is 2.3% (10s+). However, if the BB was to call considerably wider than the Nash range here (perhaps no knowledge on ICM or not wishing to be pushed around), let’s say with 25% of hands then pushing the Nash range will be unprofitable and as a result of being called 1 in 4 times the range the SB can profitably push drops from any two cards to just 34%!

On the reverse, there will be situations where you can profitably push wider than Nash when your opponent is calling tighter so for the most part playing Nash poker will be far from optimal. The most profitable thing you can do is adjust to your opponents ranges, whether that means pushing wider than Nash, pushing tighter than Nash, or perhaps not pushing at all.

For instance, consider these situations:

  1. It’s the first hand of a SNG at 10/20 and you’re dealt AA in the CO with 3 players behind you. It’s an unexploitable strategy to push all-in for 75bbs but far from optimal for obvious reasons, and you can do much better by making a standard 2.5x-3x raise.
  2. You are heads-up against a tight opponent in a SNG with equal stacks (4.5k/4.5k) and blinds have reached 150/300. A lot of players go into auto-pilot and start pushing Nash since the blinds are high. However, with more thought, you deduce your opponent is timidly only calling the top 10% of hands and almost never flatting out of position. Indeed, pushing Nash against this type of opponent will be profitable but it will be far more profitable to push even wider i.e. any two in this spot since your opponent is calling much tighter than Nash and you can expect to take down the high blinds so often.
  3. Elaborating on the second situation, if you deduce your opponent isn’t re-shipping wider than his all-in call range (10%), it will then be undoubtedly more +EV to raise most of your holdings as opposed to shove. This means you can never expect to get exploited and will save you chips the times you run into the top 10% of their range and you can be quite happy about raise-folding the times they do re-ship. Remember, just because a hand is +EV to shove doesn’t mean that is the most profitable way to play it and Nash isn’t able to account for the EV of raising or limping against shoving.

Additional spots to deviate

Nash also fails to consider the skill level of opponents and their tendencies. Consider this example: you’re on the BTN in a 6max bubble 100/200 blinds with a 1k stack and both your opponents have 4k each. Nash assumes the SB will push any two and the BB will call very tight. However, it’s more than likely the SB won’t push any two and instead wait for the short stack to bust or at least be at risk. This means if stacks very rarely end up all-in the times you fold, you can push wider than Nash so long as you aren’t getting called too wide. Now, on the other hand, if your opponents in the blinds have a lot of beef with each other and/or a poor understanding of ICM i.e. so that the times you folded, the SB would automatically shove very wide and the BB with no regard of ICM/not wishing to be outplayed called very wide then the suggested range for you to push will be super tight – way tighter than Nash, even as the short stack. This is because the times you fold the other opponents end up all-in a high % of the time and this is excellent for your equity since bar an unlikely split you’ll coast into the money which is great since you’re clear short stacked on the bubble with not a lot of equity.

Conclusions

Summing up, in spite of Nash’s limitations, it’s an excellent starting point for beginners who will begin to understand typical ranges and what they can generally push in certain spots before constructing their own ranges based on their situation reading skills and knowledge/reads of opponents. For this reason, the Nash calculator is a very useful feature, although it’s important to keep in mind some of the limitations that come with Nash:

  • Assumes play is limited to push-fold; therefore unable to measure EV of limping or raising.
  • Assumes opponents not only have knowledge of Nash but are following a set of rules – very rarely is this the case.
  • Ignores opponents skill levels and tendencies – for this reason, it can make sense to push wider or tighter than Nash based on the situation.
  • Assumes opponents shove top hands in their range excessively when they are much likelier to open raise. This weakens their pushing range and leaves players unbalanced in spots.

MikeyMcD45

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