ICM Calculations

I recently played in a $4 180-man where an interesting ICM decision came up.  ICM (Independent Chip Model) is a complicated concept that takes into account the sliding pay scale of tournament poker.  The tournament winner ends up with all of the chips in play, but not all of the money in the prizepool.  So the value of each chip in your stack diminishes as your stack grows (and vice versa).  This can lead to some complicated decisions at the money bubble, at a pay jump, and especially on the final table.

In this situation, 8 players remained on the final table. Blinds are 500/1000 ante 100.  Payouts are as follows:

1st - $216.00
2nd - $144.00
3rd - $85.68
4th - $57.60
5th - $46.80
6th - $36.00
7th - $25.20
8th - $18.72


Chipstacks (after ante and blind posting)
Hero      - 52,674
Villain    - 49,102
Player 3 - 34,447
Player 4 - 33,480
Player 5 - 31,653
Player 6 - 31,310
Player 7 - 20,798
Player 8 - 14,236

Hero is dealt A♣ K♠ and raises to 3,000.  The Villain raises all-in to 49,102.  What should the Hero do?

Chipstacks and ICM equity (if Hero folds)

Hero      - 49,674 - $98.97
Villain    - 54,402
Player 3 - 34,447
Player 4 - 33,480
Player 5 - 31,653
Player 6 - 31,310
Player 7 - 20,798
Player 8 - 14,236

Chipstacks and ICM equity (if Hero calls and wins)

Hero      - 104,076 - $145.37
Villain    - 0
Player 3 - 34,447
Player 4 - 33,480
Player 5 - 31,653
Player 6 - 31,310
Player 7 - 20,798
Player 8 - 14,236

Chipstacks and ICM equity (if Hero calls and loses)

Hero      - 3,572     - $28.95
Villain    - 100,504
Player 3 - 34,447
Player 4 - 33,480
Player 5 - 31,653
Player 6 - 31,310
Player 7 - 20,798
Player 8 - 14,236

Notice that if we call and win, we gain $46.40 in equity over folding.
However, if we call and lose, we lose $70.02 in equity.

To determine if we should call or fold, we need to multiply our winning chances against our opponent's range by our equity of winning and add that to our chance of losing multiplied by our equity of losing.  Then we compare this to our equity of folding, which is $98.97.

Some mathematical experimentation yields the breakeven point to be 60.15% here.  So if we are >60.15% against the villain's range, we should call and if we are <60.15% we should fold.
(.6015*$145.37) + (.3985*$28.95) =  $87.44 + $11.54 = $98.98

We are rarely going to be >60.15% against a hand range, holding A-K.  If the villain were to shove any pair and any broadway hand (hand where both cards are A, K, Q, J, or 10) we would be 58.84% and we should fold.  We should even fold if he somehow were to show us J-10 suited since we are only 59.49% to win!

However, keep in mind that this is a very specific scenario regarding the two chipleaders on the final table.  If we switch the Hero to the 7th place stack, things change dramatically.

Chipstacks and ICM equity (if Hero folds)

Hero      - 17,798 - $56.53
Villain    - 54,402
Player 3 - 52,674
Player 4 - 34,447
Player 5 - 33,480
Player 6 - 31,653
Player 7 - 31,310
Player 8 - 14,236

Chipstacks and ICM equity (if Hero calls and wins)

Hero      - 43,896 - $92.28
Villain    - 28,304
Player 3 - 52,674
Player 4 - 34,447
Player 5 - 33,480
Player 6 - 31,653
Player 7 - 31,310
Player 8 - 14,236

Chipstacks and ICM equity (if Hero calls and loses)

Hero      - 0            - $18.72
Villain    - 17,449
Player 3 - 52,674
Player 4 - 34,447
Player 5 - 33,480
Player 6 - 31,653
Player 7 - 31,310
Player 8 - 14,236


Now if we call and win, we gain $35.75 in equity
If we call and lose, we lose $37.31 in equity

(.514*$92.28) + (.486*$18.72) = $47.43 + $9.10 = $56.53 which is our folding equity

Now our breakeven point is 51.4%.  Even now, you must be able to include A-J offsuit in the villain's range for calling to show a longterm profit (or it will profit if you add A-Q but remove A-A, thinking that the villain will 3-bet aces w/out shoving it).