# Nash's ICM calculation

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.

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)

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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It seems to me that you're conflating three separate concepts:

• ICM equities
• EV calculations
• Nash equilibria

I'll go over these in turn an explain how push/fold calculator software uses them to produce solutions for real-life scenarios.

## ICM Equity

ICM equities are a way to determine each player's equity share in poker games with non-linear payout structures (i.e. tournaments). ICM equities are expressed as a percentage of the total prize pool and are calculated solely based on the distribution of chips among the players and the tournament payout table.

For example, in a typical 6-max hyperturbo SNG (payouts 65%/35% for first and second finisher respectively), with 3 players remaining whose stacks are:

• player 1: 1500 chips
• player 2: 1000 chips
• player 3: 500 chips

an ICM algorithm might tell you that the first player's ICM equity is approximately 45% of the total prize pool.

I've made a video explaining how different algorithms calculate these equities here: https://www.youtube.com/watch?v=JtBAxjR21o8 and you can use our (https://omni.poker) ICM equity calculator for free if you want to play around with these (instructional video here: https://www.youtube.com/watch?v=XuEBxON9fGQ).

You can read up on ICM algorithms in the Mathematics of Poker by Bill Chen. Ben Robert's algorithm is described here: http://www.pdf-archive.com/2014/08/04/benrobertsequitymodel/benrobertsequitymodel.pdf.

ICM equities are used in practice to:

• negotiate final table deals (when the last few remaining players in an MTT decide to chop the remaining prize pool, they'll usually allocate prize money according to the ICM equities implied by their chip stacks)
• adapting the concept of pot odds to tournament situations (called tournament odds and described in the book Kill Everyone by Nelson/Streib/Heston)
• as a heuristic when doing EV calculations in tournament settings

## EV Calculations

To do an EV calculation simply means taking a simplified (in terms of game tree size) poker game, intended to model some real poker game with reasonable accuracy, writing down the full game tree for the simplified game, and then calculating the expected value of each decision in order to be able to pick the best decision.

For example, arguably the simplest useful model of preflop poker is the push/fold game. In this game, one player may go all-in or fold and the other player(s) may call the all-in or fold.

Continuing our scenario above, let's say the first player folded and the second player must now decide between pushing and folding. We get a game tree that looks like this:

``````             p1_fold
/       \
p2_push   p2_fold
/       \
p3_call   p3_fold
/   \
p2_win  p2_lose
``````

We now need to assign values to each choice node, and this is where ICM equities come into play.

Let's take the simpler case first. If p2 decides to fold, what is his expected value? One way to answer this question is to look at how ICM equities are affected by a given choice. To make this concrete, let's say the current blinds are 20/40, so that when player 2 folds, 20 of his chips will move to player 3.

I plugged these values into an ICM equity calculator and got the following results (using the Malmuth-Weitzman algorithm):

• before the hand: p2 has 35.67% ICM equity
• after the hand: p2 has 35.06% ICM equity

so the EV of folding is -0.61% ICM equity

What's the EV of player 2 going all-in? This is considerably more complex. First off, we need to model player 3's behavior, i.e. we need to estimate the distribution of hands that player 3 will call an all-in with. Given this information, we can then calculate a conditional probability (conditioned on player 2's holdings' card removal effects) of player 3 calling an all-in as well as the equity of player 2's holding against player 3's range.

We'll end up with something like this:

``````EV_chips(push)     = EV_chips(p3_calls) + EV_chips(p3_folds)
EV_chips(p3_calls) = P(p3_has_calling_hand) * (P(p2_wins)*500 + P(p2_loses)*-500)
EV_chips(p3_folds) = 1-P(p3_has_calling_hand) * 40

where P() indicates the probability of an event
``````

This gives us the Chip EV (i.e. the expected number of chips won or lost) of player 2's decision to push all-in:

• the 500/-500 in EV(p3_calls) are simply player 3's stack size
• the +40 in EV(p3_folds) is the big blind

which we can convert to ICM EV by plugging the updated chip stacks into an ICM equity calculator in each case.

Commercial push/fold calculators such as our PUSH/FOLD module or ICMIZER do these steps for you, i.e.:

• you input an opponent model (a calling range for each opponent)
• the calculator builds a game tree, assigns probabilities, calculates chip EV and converts that to ICM EV
• the calculator does this for every starting hand and shows you the hands with higher EV(push) than EV(fold) as your pushing range

## Nash Equilibria

Finally, let's talk about Nash equilibria in poker.

A Nash equilibrium is a solution concept for two-player zero-sum games which describes a strategy pair (i.e. one strategy for each player) such that no player can unilaterally improve his EV by deviating from the solution strategy.

What does that mean in practice? Recall from the EV calculation example above that we had to model our opponents behavior in order to do our calculation. While we may have a good idea of how specific opponents or populations of opponents behave (and we may be able to base our modeling on a large database of hand that we've played) we might instead want to know what optimal play would look like against an optimal opponent.

To determine optimal play, we'll essentially solve a given game tree many times, with players adjusting to each other in an alternating fashion.

E.g. to solve our game tree above, we'd start by putting in any random range of calling hands for player 3 and calculate the optimal range of pushing hands for player 2. We'd then keep this range of pushing hands fixed and solve the tree again from the perspective of player 3 (i.e. calculate the optimal range of calling hands given player 2's pushing range). We'd now go back to looking at the tree from player 2's perspective and calculate the optimal pushing range given player 3's new calling range, and so forth.

This process will eventually reach a stable state, where neither player can improve his expectation by changing his range and this is the game's Nash equilibrium.

For the curious, the algorithm I just described is called Fictitious Play and is only minimally more complex in practice than what I outlined above (rather than always switching to the best response strategy one mixes the best response strategy into the current strategy to avoid oscillation and rather than running it until it has reached a stable state, one measures the distance of the current solution to the true solution, by measuring the magnitude of EV changes, and stops once it's close enough).

Why are equilibrium solutions interesting? IMO there are three main uses:

• to get a feel for what correct play looks like in a given situation, which one can use as a baseline for one's strategy development
• as a default strategy against unknown or scary opponents
• at the highest level of play today (e.g. high-stakes HUSNGs) players employ optimal or near-optimal strategies in many situations and it's necessary to play GTO strategies oneself in order to compete

As a final note, you may have noticed that I used two-player games for my examples. This is because only in two player games do Nash strategies give a non-negative profit guarantee. Informally, this is because any game has some value and when there's only two players, one player foregoing some of that value must mean that the other player captures it. In multi-player games, one player making a -EV decision might mean that some or all of his opponents end up capturing that value which makes playing equilibrium strategies in that setting a lot less appealing: you may be playing optimally against an opponent, but him playing sub-optimally against you may benefit another opponent rather than yourself.

HTH

• The notion of Nash equilibrium is defined for any type of game, not just 2-player zero-sum games, with any number of players, even a continuum of players.
– plm
Commented Dec 15, 2023 at 3:16

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.

• 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. Commented Jan 7, 2013 at 9:50

essentially what the Nash solution tells you: How to play without giving the opponent an opportunity to exploit me So in tournament poker, what the Nash chart will tell you is how to play each hand depending on the number of chips you have, your cards and how many opponents are left to act. To arrive at the solution, the algorithm iterates until there is no more adaptive moves left. I hope that makes sense :)