The buttons **Calculate Nash** and **Calculate** do two very different things.

1) **Calculate Nash** finds the approximate Nash solution for *all* players.

It calculates and displays all ranges with the goal to minimize the potential size of the exploit (by the definition of the Nash equilibrium).

The precise Nash almost never exists. A precise Nash solution would mean that there is no area for an exploitation for any of the players. It would represent a completely static solution and every exploit for each player will be precisely 0, where no player indeed is interested in deviating from the Nash strategy at all.

Currently, ICMIZER is using pure strategy when it performs calculations and finds Nash solution. In pure strategy, we are either playing (100%) or folding (0%) a hand.

In ICMIZER 3 we will add mixed strategies, where each hand can be played some % of the time, and folded the other, with values falling between 0% and 100%.

Because of this additional flexibility, mixed strategies allow finding more stable Nash solutions where the potential exploit size is smaller.

If the potential exploit is smaller, the jump when you perform a calculate will also be smaller.

2) **Calculate **finds the best response range for just one specific hero position, which exploits all other players strategies to the maximum.

It fixes all opponent ranges right where they are and calculates one range for the hero in the current situation.

Since Nash equilibrium ranges where calculated approximately, this precise result will almost always be somewhat different. In some special cases, it can be significantly different, which would indicate that the situation is special and potentially isn't best solved with a push/fold approach.

Some details about these types of situations can be found in this article.

So for the **Nash calculation**, only the stacks, bets and the tournament are the input, and** all the ranges (the strategy for all players)** are the result of the calculation.

For the **Calculation**, however, stacks, bets and the tournament are the input, but on top of that all relevant opponent ranges are also the input, and the result is just a **single **range.

Due to the approximate nature of the Nash equilibrium solution, there almost always is some room for exploit

and the range changes from initial range that was part of Nash solution after the **Calculate **is performed on it.

When we are talking about Calculate Nash **Beta** functionality, which is currently used in situations with 1 or 2 non-all-in raisers, the approximate solutions usually have more space for potential exploit compared to Nash approximations without non-all-in raisers (Calculate Nash without the Beta). Because of this, the % jump can be pretty significant, as you are observing on the screenshot.

The only way to minimize this jump significantly is to introduce mixed strategies, which we will do in ICMIZER 3.

However, there is one caveat with mixed strategies. While Nash equilibrium with mixed strategies is more stable, has less area for exploitation and because of that

has a smaller effect of what you are seeing on the screenshot ("ranges jump less"), this stability comes with a price. And this price is that most ranges of the Nash solution will have 1 or

usually much more hands with random percentages, where say one hand is played 57% of the time, the other is played 73% of the time etc, etc, etc.

So while theoretically correct, such strategy could only realistically be followed by a bot, and the real players usually have a strategy which is much closer to the pure strategy.

Where they either play a hand or fold it and do not roll a d100 dice to determine a plan with a certain hand according to a complex table.

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