Tuesday, January 13, 2009

ICM PLO8 article

ICM Decisions in Split-Pot Tournaments

The Problem

Congratulations, we have just made it to the final table of a Pot-Limit Omaha Eight or Better Multi-Table Tournament! Now what?

There are 8 remaining at the final table. We are past the bubble and playing to maximize our value on a sliding pay scale. The Payouts, Chip Counts, and Current ICM Values (aka cEV) prior to posting our big blind are as follows:

Position Payout
1 $1,950
2 $1,475
3 $1,100
4 $840
5 $575
6 $450
7 $350
Total $6,740

Chips Current ICM Value
Player 1 21,000 $936
Player 2 7,000 $601
Player 3 38,000 $1,181
Player 4 6,000 $568
Hero 26,000 $1,020
Player 6 32,000 $1,107
Player 7 (BS) 53,000 $1,327
Total 183,000 $6,740

Player 7 has been working the big stack and raising the table more than her share. In the last 10 hands, she has been involved in 6 hands and her cards were exposed four times at showdown. Those hands were:

A578 – Three of one suit
A8JJ – Three of one suit
QQ89 - rainbow
AKJ9 - double-suited

The blinds are 1000/2000 and you are in the BB. As expected, the Bully raises to 5000, and the table folds around to you. If find you have a very strong hand, what is the best possible play? Let’s take a look our table observations and then take a look at the math.

• There are two stacks at less than 3M, in serious danger of elimination
• The difference between being eliminated now and outlasting the two small stacks $575 compared with $350, or a 60% improvement in pay.
• A double-up here makes us the chip leader, giving us a better shot at first prize, especially as cautious as this table has been.
• Based on prior behavior, if we reraise, we expect that the bully will reraise and put us all-in 100% of the time.
• Based on prior behavior, if we flat call, we expect villain to respond to our flop bet with an all-in move about 70% of the time and to fold the other 30%.

Now let’s look at the math. Here are our ICM values in the following scenarios.
• We fold preflop ($995)
• We flat call and fold on the flop ($948)
• We bet the flop and bully folds ($1079)
• We chop the pot and profit by T500 ($1031)
• We double up ($1329)
• We are eliminated ($350)
• We get ¼ of the pot ($803)
• We get ¾ of the pot ($1195)

Option 1 - Push or Fold Only

Here is the ICM value of AA23 double-suited, the strongest possible hand in PLO8. We will assume that villain repots and put us all-in, and we will use propokertools.com simulations of AA23ds versus a random hand to determine the remaining distribution.

Percent ICM Value Result
Losses 17.50% $350 $61.25
Chops 18.35% $1,031 $189.14
Scoops 61.41% $1,329 $816.18
3/4 1.71% $1,195 $20.40
1/4 1.04% $803 $8.32
100.00% ICM Value $1,095

Compared to folding AA23 double-suited preflop ($995), we confirm that raising with the best possible hand is a positive equity play and increases our cEV by about 10%. The ICM model needs to be very severe before we should contemplate folding a monster starting hand.

If we assumed that villain would fold pre-flop 30% of the time that we reraise, the numbers for AA23 would come out as follows.

Percent ICM Value Result
villain folds preflop 30.00% $1,079 $323.70
Losses 12.25% $350 $42.87
Chops 12.84% $1,031 $132.40
Scoops 42.99% $1,329 $571.32
3/4 1.19% $1,195 $14.28
1/4 0.73% $803 $5.82
100.00% ICM Value $1,090

Assuming that villain will always call our reraise, the weakest double-suited hands we can push with and still be marginally profitable include A2Q5 ($996), A35J ($994), A45J ($996), A5JJ ($996). Any AAxx hand that includes one low card will be profitable, but the strong high-side hand AAKK double-suited is only worth $970, somewhat worse than folding that hand preflop.

For the remainder of the article, we will focus on the hand A23K double-suited. Its pushing value is $1009. If the villain folds 30% of the time when we reraise with this hand, our value goes up to $1030. Our risk of ruin is 24% if villain always calls and 16.5% if villain folds 30% of the time that we repot.

Option 2 – See and evaluate a flop and push a wide range of draws and made hands

A second option is to see and evaluate a flop, and only push where we have a flop that meets any of the following conditions.

1) We have a made nut low
2) We have a nut-low draw with one card needed (treat this as a semi-bluff)
3) We have a made high side hand, such as trips or better.
4) We have a high side only semi-bluff hand (Flush draw, pair of aces, pair of kings, and we will classify two pair as a semi-bluff for outcome calculation purposes)

First we generate the number of flops that qualify, and determine the number of flops where we will plan to check fold. For A23K double-suited, we estimate 5840 combinations out of 17, 296 possible flops (34% chance) where we have a flopped nut low or a nut low draw. In addition to these low chance flops, these high-side flops exist outside of the low flop chances.

High Hand Flops Combinations Percentage
Trips or full house 552 3.19%
Quads 4 0.02%
Broadway Straights 64 0.37%
Flopped Flushes discounting 3 low card flushes and 2 low card flushes 140 0.81%
Flush draws with 2 high flush cards and any third card 592 3.42%
Flush draws with 1 high flush cards, one low flush card and one other high card 840 4.86%
Two Pair plus one high card 864 5.00%
(discount flush draw combinations) -272 -1.57%
A King plus one or two high cards (two pair draw) 1752 10.13%
(discount flush and flush draw combinations) -72 -0.42%
An Ace plus two high cards (two pair draw) 513 2.97%
(discount flush and flush draw combinations) -98 -0.57%
Total 4879 28.21%

We estimate that we can push on 62% of all possible flops. We will check-fold the 38% of flops that do not contain a low draw or at least a pair of kings. We simplify the model with the following assumptions.

• Villain will fold 30% of the time that we pot the flop
• For any made low flop we are mostly free-rolling for the high, and chopping 75%, scooping 25%.
• For any two-card low flop, we are struggling to survive when called, and we are losing 40%, chopping 50%, and scooping 10%.
• For any high-hand flop with a made hand, we are scooping 60%, chopping 30% and losing 10%.
• For any high hand flop with a draw or any two-pair and one-pair hands, we are losing 40%, chopping 30% and scooping 30%.

Our outcomes table appears as follows

Occurrence Combinations Percentage
Fold Flop 6577 38.03%
Made Low Chop (75%) 2122.5 12.27%
Made Low Scoop (25%) 707.5 4.09%
Low Draw Loss (40%) 1204 6.96%
Low Draw Chop (50%) 1505 8.70%
Low Draw Scoop (10%) 301 1.74%
Made High Scoop (60%) 456 2.64%
Made High Chop (30%) 228 1.32%
Made High Loss (10%) 76 0.44%
High Draw Loss (40%) 1647.6 9.53%
High Draw Chop (30%) 1235.7 7.14%
High Draw Scoop (30%) 1235.7 7.14%
Total 17296 1

We reorganize our outcomes as follows.

Occurrence Percentage ICM Gross Value ICM Calculated
Fold Flop 38.03% 948 $360
Villain Folds Flop to our bet (30%) 18.59% 1079 $201
Scoop 10.93% 1329 $145
Chop 20.60% 1031 $212
Loss 11.85% 350 $41
Total 1 $960

Our expected cEV from a stop and go where we push the best 62% of flops is actually worse than folding preflop. Our risk of ruin drops from 24% to 12% compared with pushing preflop.

Option 3 – See and evaluate a flop and push a narrow range made hands

What if we evaluate a flop and only continue on hands where we have a very strong flop? Let us revise our betting criteria as follows:

1. We have a made nut low
2. We have a made high hand, two pair or better.
3. We have a nut low draw (need one card to make low) plus any flush draw or open ended low straight draw.

Let us assume that villain will fold 30% of the time that we push on the flop. Let’s also assume that we are mostly either scooping or chopping when we push on the flop, and that we will include a small portion of losses rather than attempting to quantify the number of times that we will get quartered. This strategy offers the lowest risk of ruin (5%) for any option, and offers the best cEV for any of the post-flop options.

Occurrence Percentage ICM Gross Value ICM Calculated
Fold Flop 52.90% 948 $501.46
Villain Folds Flop to our bet (30%) 14.13% 1079 $152.47
Scoop 11.93% 1329 $158.50
Chop 16.14% 1031 $166.35
Loss 4.91% 350 $17.19
Total 1 4737 $996

In other words, compared to folding pre-flop, it is marginally profitable to call with A23K double-suited and only push on highly selective criteria where we have a virtually guaranteed chop and are free-rolling for the other half of the pot. Despite offering a lower overall value than pushing preflop, this may be the best option when playing against certain categories of villains, because it offers the lowest risk of ruin (5%) for any option, and offers the best cEV for any of the post-flop options.

Option 4 – Pure Stop and Go

It can be argued that the villain will tighten up if we put her to the test. If we treat this as a stop-and-go play where we always bet the flop, and villain folds to 65% of our all-in flop bets (an optimistic scenario, unless villains standards for continuing are much tighter than our own) and puts us all-in on the remaining 35% of flops.

Occurrence Percentage ICM Gross Value ICM Calculated
Fold Flop 0.00% 948 $0
Villain Folds Flop to our bet (65%) 65.00% 1079 $701
Scoop 5.46% 1329 $73
Chop 10.30% 1031 $106
Loss 19.23% 350 $67
Total 1 $948

The stop and go scenario above is close in value to the call-and-evaluate model in option 2. However, the number is ‘terribly optimistic.’ If we change our assumption so that villain would fold to our stop and go in the same proportion as we folded in option #2 (38 %), the outcome would degrade as follows.

Occurrence Percentage ICM Gross Value ICM Calculated
Fold Flop 0.00% 948 $0
Villain Folds Flop to our bet (38%) 38.00% 1079 $410
Scoop 9.68% 1329 $129
Chop 18.25% 1031 $188
Loss 34.07% 350 $119
Total 1 $846

It can be argued that if the villain is only folding 37% of the time, villain will be on a draw a significant portion of the time. When we arbitrarily increase the overall chops and scoops, the numbers look better, but this is still not a profitable play compared with pushing only on the strongest flops.

Occurrence Percentage ICM Gross Value ICM Calculated
Fold Flop 0.00% 948 $0
Villain Folds Flop to our bet (38%) 38.00% 1079 $410
Scoop 14.00% 1329 $186
Chop 26.00% 1031 $268
Loss 22.00% 350 $77
Total 100.00% $941

Concluding Thoughts

Much like any other MTT, there are certain times in the tournament when it is correct to push with virtually any hand, and there are times when we should only consider pushing our strongest hands. In this situation with two life-support stacks and a villain who does not perform hand valuations well, we must be highly selective on our pushing hands, since we have minimal fold equity and we stand to lose so much more value in elimination than we could gain in a double up.

By the same token, we should not necessarily cave in to a big stack bully with our strong but not monster hands: with the chips to see a flop without committing, we may prefer to see a flop with our stronger two-way hands, and push on flops where we are very likely to survive and preferably thrive, but give up on flops that do not give us much hope in either direction, without giving up too much tournament equity. Because it reduces volatility, this strategy is particularly recommended when you believe you have a significant skill edge over the villain.