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# Calculating equilibrium ranges for raise situations (Update ICMIZER v2.9)

## Calculating equilibrium ranges for raise situations

You’ve asked us to make it work and we have. In ICMIZER 2.9, you can finally calculate Nash equilibrium ranges for raises of any size!

In game theory, a Nash equilibrium is a set of strategies such that no player is able to gain an advantage by changing his/her strategy as long as other players don’t change theirs. Poker pros rely on this kind of optimal strategy to build their success. By playing Nash, they remain as unexploitable by opponents as possible. Of course, an experienced pro knows when to stray from Nash to effectively take advantage of flaws and shortcomings in opponents’ strategies. But when playing against unfamiliar opponents, or very strong ones, Nash is the norm.

In the five years ICMIZER poker calculator has been on the market, it has supported only all-in situations, both for the main branch and for subtrees.

Clearly, there exist certain ratios of blinds and effective stack size for which you’d not want to go all-in. This is why it’s so important to be able to process smaller raises. Even at depths that are more common for the push-fold phase, calculating an open-raise situation is also necessary: a raise may be made by your opponent, and of course we can’t do anything about that.

By adding raise calculations, ICMIZER has significantly expanded its use cases. Before v2.9, if you wanted to calculate the EV of a raise smaller than an all-in, or the EV of a push against a raise, you had to manually add all ranges for all players still in the hand. Now the application does this for you automatically, and even better, it calculates the Nash ranges right away.

As a response to a given raise amount, ICMIZER 2.9 automatically calculates the ranges of all-in re-steals for all other players. However, for a scenario when a 3-bet is made on a less than full stack, you can manually enter the re-raise amount and have the application identify the equilibrium strategy accordingly.

Let’s look at some sample scenarios.

### Scenario 1

It’s early in a SNG 6-max hyper tournament, we’re on the small blind and the button makes a mini-raise.

By finding optimal ranges, we can learn a lot of useful things. For the button, we see their optimal raising range is 40%. If they open too wide, they will have to fold too often to all-ins from the blinds. That reduces their EV from that position with a wider raising range.

If a strong player makes a wider re-steal from the button, for example with a range of 55%, then you know they will call with a tight range. You can then go all-in with a pretty wide range and thus punish them for trying to exploit tight play from the blinds.

If the raise is, say, 30%, then you know the player isn’t that strong and is playing tighter than necessary. You stand to beat them on the flop where their probably play in a rather straightforward manner. Still, don’t go all-in on them on the preflop, as they will tend to have a good enough hand to call with. In this case you should be glad they’re under-attacking the blinds and losing out by folding 10% more often than necessary. But when this player does enter the pot, you should restrain yourself and play the preflop without doing anything fancy.

### Scenario 2

We’re at the final table of a major tournament with a less than lucrative payout structure. Hero is on the BB with an average-size stack. The cutoff who has a little over 11 BBs goes all-in. The small blind, the chip leader, isolates the cutoff with their own all-in. The overcalling range for Hero is QQ+, so Hero folds JJ.

Let’s tweak the conditions: the short stack opens with a 200 raise and the chip leader goes all-in. How will this affect our decision?

As it turns out, this makes a huge difference! The chip leader can re-raise the cutoff a lot more widely, and this expands our calling range to 99+ and АК. Pushing with JJ gives us an increase of 0.28% tournament equity.

### Scenario 3

In a standard Spin & Go tournament, the button goes all-in in the first hand and the small blind folds. Hero’s calling range on the BB is 66+, AT+ or KQs.

Now let’s calculate the optimal strategy for raising 3 BBs.

ICMIZER suggests the button should open with a 3-BB raise in 32% of hands. In response, Hero can push with 19% of hands, i.e. 44+, A2s+, A7o+, K9s+, KTo+ or QTs+. The button must defend in the 14% range with 33+, A7s+, A8o+, KJs+ or KQo.

### Scenario 4

Let’s look at a hand from the final WCOOP table in which Ukrainian regular Roman “RomeoPro” Romanovsky made a super-tight fold after an all-in from the UTG.

According to ICMIZER, Roman should call with TT+ or AK. Explaining his decision to fold, RomeoPro said his opponent was quite tight and never goes all-in as widely as our application advises (44+, A8s+, A9o+ or KJs+). Let’s see what ICMIZER suggests in response to a standard raise. Here’s the recommended push range and how it changes the tournament equity:

An all-in with AKs gives the hijack +0.13% tournament equity.

Interestingly, calling the all-in with AKs would give +0.25% – almost twice as much! OF course, this is valid only against the optimal Nash equilibrium strategy.

## Why is ICMIZER 2.9 in beta?

We’ve added this functionality as a beta to enable our customers to calculate Nash equilibrium for raises of any size. ICMIZER currently uses linear methods to identify Nash equilibriums. This means that player ranges are chosen from just 169 possible options and applies a hand rating table (in order of descending hand strength), as opposed to all the possible ranges. The equilibrium strategies it finds are therefore approximate. When we deliver the final release, it will feature exhaustive search, not limited by the 169 pre-set range options.

In practice, for most situations the strategies this beta version finds are very similar to those that can be called ultimately accurate.

In addition, ICMIZER calculations currently do not consider mixed strategies, only pure ones: either a hand is included in a range or it isn’t. In other words, the same hand can’t be played with a 40% fold and a 60% raise. In the final release we’ll achieve full Nash equilibrium for mixed strategies. We’re really focusing on making this happen. Some could say that a full solution for mixed strategies is overkill, that the surgical precision it provides is of no use in real poker. Still, having this ultimate accuracy available will give a peace of mind to those players who strive for perfection.

## Locking and silencing

One intrinsic limitation of an equilibrium strategy is that all the ICM calculators on the market today are able to analyze all-ins for a maximum of three players. Multiple all-ins from four or more tournament players push the computational demands so high that such solutions are inaccessible at this time.

Consider the following situation.

We’re on the cutoff and looking at a raise followed by an all-in. ICMIZER calculates our overcalling range based on the assumption that entering this hand for us ends all the action – no further overcalls are possible, except of course for the UTG. Even if the BB has hole aces, they will fold them.

According to the calculations, the weakest off-suited hand that we can enter the pot with is AK. But how much this conclusion differs from the full solution of this problem, which no commercially available ICM calculators can yet deliver, remains to be seen.

For the same reason, it is quite inconvenient to calculate situations where a player automatically goes all-in after the blinds and ante are posted. For the players at the table they’re as good as absent, but for the calculator they’re not. This complicates things, a lot.

This is why in ICMIZER 2.9 we’re forced to ‘lock’ some players and ‘silence’ them, i.e. strip them of their vote in the hand.

On this screenshot we see that we can calculate the all-in range for Hero on the cutoff, but if the button decides to enter the hand, because of the player number limitation the UTG will have to auto-fold. That’s why in this situation we ‘silence’ the button and the blinds to protect the player who’s already entered the pot (so-called VPIP-protection).

Hero’s all-in range is 99+, AQs+ and AK. UTG can call two all-ins with TT+, AQs or AK.

### Why the Calculate button is sometimes unavailable

For the same reason described above, namely that multiple all-ins are limited to three, sometimes we can calculate Nash equilibriums but don’t have access to the Calculate button.

In this example we see a raise and two all-ins made before us. We are unable to calculate Hero’s response to this with the Calculate button.

The button just above it which reads “Calculate Nash Equilibrium Beta” will identify ranges that approximate the Nash ranges for all players. We use the Calculate button to determine the optimal response for a given player. Obviously, when a player is locked we cannot calculate anything for them. Meanwhile, strategies for silenced players can be studied freely.

## What's next?

With this long-awaited update, our subscribers can finally calculate Nash equilibriums for raises, but we’re not stopping there. After these new optimal strategy calculations go from beta to final release, which should happen soon, our team will focus on a more ambitious goal. We will work to bring the ICMIZER engine to its logical conclusion and teach it to quickly and accurately identify full Nash equilibriums for mixed strategies. Of course, the final release will retain the ability to consider FGS up to depth 6.

We’re sure this plan can be realized this year. ICMIZER 3.0 will, for all intents and purposes, become the pinnacle of the ICMIZER engine. Further development will be aimed at enhancing its usability and user experience. In particular, we’ll refine hand history analysis and continue to work on SNG Coach so it can train you to counter raises. And naturally we’ll continue to listen to our customers who are constantly using ICMIZER and have first-hand knowledge of what is missing and what should be improved as a priority.