by: Jason Waddell

Poker is a hard game. Tournaments payouts add a layer of complexity to the process. Chips won do not directly correlate to dollars earned. Let’s take a relatively simple hand.

We are in the top 3 of a tournament with the following payout structure:

- 1st: $30.15
- 2nd: $18.10
- 3rd: $12.05

The player on the button folds, and we are left with a rather binary decision: push or fold. Although our hand is terrible, the cards are live, and letting the small stack double-up for free by taking the blinds isn’t a terribly attracting. How can this hand play out? Discretely there are 4 possibilities:

- We raise, are called and win.
- We raise, are called and lose.
- We raise and our opponent folds.
- We fold.

How do we make that decision? Enter the Independent Chip Model (ICM), a model which translates the players’ tournament holdings to their expected tournament winnings in dollars (or any other currency). Basically, plug in the payout structure and each player’s stack size, and ICM gives you each player’s expected winnings.

The above four outcomes result in the following stack sizes at end of hand:

Starting Stacks:

1682 (Button)

1790 (Hero)

1028 (BB)Stacks if we raise, are called and we win:

1652

2848

0Stacks if we raise, are called and we lose:

1652

762

2086Stacks if we raise and villain folds:

1652

2150

698Stacks if we fold:

1652

1610

1238

In terms of ICM for the hero (that’s us!), we have:

ICM pre-hand: 21.4

- ICM if called and we win: 25.6 (+4.2)
- ICM if called and we lose: 16.6 (-4.8)
- ICM if raise and villain folds: 22.90 (+1.5)
- ICM if we fold: 20.6 (-0.8)

Broken down like this, the decision is purely mathematical, as a function of two quantities:

- The probability villain folds to our shove
- The probability that we win, given a call.

This is a simple expected value calculation, bread and butter for any statistics student. Villain’s cards are unknown, and using simulation we know that the probability our hand (57 offsuit) beats a random hand is 40.5%. This is a bit of a simplification, as villain’s probability of calling could change as a function of his hole cards, but in this situation it should be fairly representative.

What is the probability villain folds? Well that’s a true unknown, and one we have to guess at. But first, let’s see what our expected ICM looks like as a function of villain’s probability of folding. The weighted expected value calculation is:

Next, we visualize this formula by examining expected ICM as a function of our villain’s probability of folding (p_fold).

If we fold, we know our expected ICM: it’s fixed at $20.6. Thus, we want to pick the option that maximizes our ICM. Here, we can calculate where the black and blue lines cross. Solving for p_fold, we get a value of 0.134, or 13.4%. So the question is, do we expect villain to fold more than 13.4% of the time?

Let’s step into his shoes for a moment. His remaining chip stack is 698. If we shove, he faces calling his stack to win a pot of 300 + 300 + 90 + 698, or 1388. This is giving him odds of almost exactly 2:1 to call, which means he is properly priced to call with basically any two cards.

Thus I expect villain to fold less than 13.4% of the time, and we go ahead and **fold our cards**.

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