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I would like to understand the Nash' ICM model and how to use it. I have found this calculators here and here, and I have tried to play with them. But I don't understand the output. Who can help me explaining me the output or telling me where i can find a good textbook for reference? I am interested in the theory too so i would like to understand the mathematics behind this numbers.

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This is pretty complex question and proper answer is worth a chapter in book (Try Collin Moshman's SNG strategy it's a must read for every sng student.)

Usage

No one ever solved Nash equlibrium for such complex game as poker, if such strategy was found, it would be unbeatable strategy. (Not most profitable though, because its static strategy and doesnt involve adjusting to our opponent play.)

We are able to calculate strategy for simplier variant of poker. In this simple variant, every player has to decide, whether he want to go all-in or fold preflop. No other moves are allowed. Because this simple variant differs so much from hold-em poker(which involves planning our strategy for multiple streets, weightening outs, counting pot odds... ), we can use this strategy only in situations, when our optimal play will likely no differ from optimal play in our all-in/fold simplified variant. Nash is therefore used just in HU when effective stacks are <12 (some authors claim its -EV and recommend playing nash when effective stacks are <8) or in SNG, usually during bubble play and all-in/fold phase. (3-5 players on 9max, 2-4 players on 6max)

I recommend reading Merseneary free e-book for more advanced and deep discussion about this topic.

Theory

I think Soner Gonul explained this well and I would just copy him.

Great tools to practice

Can be found here

Understanding output of calculators

I will refer to one instance of calculation.

First table

Informs us about structure of sit-n-go, number of players and number of runs(These tools don't calculate NASH but only estimate it pretty correctly. Runs is number of games simulated to get result.)

Second table

Player - Refers to player position at the table

Stack - Number of chips player has

push% - If all his opponents play correctly, he should push top push% percent of hands.

EQPre - This is the fair division of prizepool to players before dealing the hand based on propability, how much money are players going to win in the long run.

EQPost - This is the fair division of prizepool to player after playing one round.

You can see, that it's decreasing for SB, he is shortstack on the bubble, he has another shortstack who can survive to next round or push to him and also he has bigstack on right side who usually has no problem to call his all in. CO benefits the most from this situation, he can get into the money after this round, although he has only 2BB. BB has so much chips, that his expected amount of prize will not change dramatically.

EQdiff = EQPre - EQPost

Third table :

Row by row:

CO should push his top 4% hands...

if CO push, BU should call with QQ+

if CO push, BU calls, SB should only with AA

if CO push, BU calls, BB should only with AA

if CO push, BU folds, SB should call with JJ+

if CO push, BU folds, SB calls, BB should call with top 58%

if CO push, BU folds, SB folds, BB should call with top 58%

if CO folds, BU should push top 13,7%

if CO folds, BU push, SB should call 5,6%

if CO folds, BU push, SB call, BB should call 9.8%

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.

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I will talk about the theory part because I'm not sure about how the mathematic part works. On wikipedia page, there are some explanation about the math part but it is little bit complicated.

Nash ICM calculation is the based on Nash equilibrium theory. This theory belongs on John Forbes Nash. If you remember there was a movie called A Beautiful Mind about John Nash's life.

In these days, It is close to impossible to find the exact Poker Nash Equilibrium strategy for games with many possible strategies. Really hard because the possibilities of a games as complex as poker are vast.

From Wikipedia;

In game theory, the Nash equilibrium is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.


Stated simply, Amy and Phil are in Nash equilibrium if Amy is making the best decision she can, taking into account Phil's decision, and Phil is making the best decision he can, taking into account Amy's decision. Likewise, a group of players are in Nash equilibrium if each one is making the best decision that he or she can, taking into account the decisions of the others.

Informal definition

Informally, a set of strategies is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks himself or herself: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?"


If any player would answer Yes, then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the set of strategies is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to all other strategies in that equilibrium

If you really interested in game theories, you can read Theory of Games and Economic Behavior. I didn't read this book but it called a legendary about game theories.

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Thanx Soner, but I know the definition of Nash equilibrium. The question is about how to use ICM model in order to solve Nash equilibrium for many players on texas hold'em. –  emanuele Jan 7 '13 at 9:50
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