Wednesday, March 28, 2007

'Checking Down' Hands in SNGs - Mathematically Correct??

Going to have a look at a couple of different 'checking down' scenarios and make sure they are correct in terms of both ev and ICM. It has been a long held assumption that this is the 'correct' play - I just wanted to make sure!! (note: first one got longer then expected so will split this into 2 parts, next one to follow later in the week)

This kind of situation happens where there is a small stack all-in - 2 larger stacks refuse to bet any further on the hand in order to maximise the chances that the smally is knocked out. At the bubble this has the effect of increasing all of the other players chances of making the money...

Lets take an 'obvious' example first and take a look at the numbers behind it. 4 players left $100 prize pool and bb=200, stack sizes below.

Player A: 5000 - $ev = $36.21
Player B: 3000 - $ev - $29.62
Player C: 2500 - $ev - $27.30
Player D: 500 - #ev - $6.87

Imagine the smally is in the CO, pushes his last 500 chips in and is called by both player A and player B... they then check the flop turn and river to maximise the chances of eliminating the small stack. The pot is thus 1500... we should look at the equity calcs for the 3 possible outcomes first;

Player D wins his $ev is now $16.89 (increase of $10.02) while A = $33.57 (<$2.64) and B = 24.77 (<$4.85) - player C who was not in the hand also loses a bit going from $27.30 to $24.77 (<$2.53).

Player A wins his $ev is $39.57 while B and C each have $30.21 - everyone improved as the bubble is now burst.

But hang on - lets look at things from the viewpoint of a (correctly) selfish big-stack. Winning here is massive in terms of leverage in the money - having more than twice the stack of his opponents. It is also close in terms of $ev (losing reduces this by $2.64 while winning increases it by $3.36 - thats a 1.3/1 ratio.

Now we can look at the hands the 2 opponents might hold - pretty wide for D right? i'd certainly push any ace or king, any 2 cards 10 or above, Q8 and J9 as well as pairs here. Likewise with player B - with only 300 to enter a pot of 1200 (and the assumption that things would be checked down) he will come in with a huge range...

So from the big stacks perspective forcing player B out of the pot on the flop might be a positive expectation move after all... even more so if he figures to beat the small stack.

My thoughts here are that player B (meduim) stacked guy can not possibly call a flop shove here without a monster. He is basically risking the $24.67 in equity he gains by folding and having player D win as a minimum.

Let us be generous and say that player B has said monster 5% of the time... player A hits top pair and decides that pushing is the best option. The maths then splits into 2 - what % of the time would we expect A to beat D and what happens when B wakes up with trips (for example).

So 95% of the time player B folds and the showdown is between A and D - with top pair for A and such a wide range for D we will give player A 75% winning chances.... no problem here $ev wise.

5% of the time player B wakes up with a monster and wins the pot. Now we get to the key factor from the perspective of the big stack... how often would player D (the small stack) win the main pot here leaving the (bigger) side pot for B. With such a range the answer is no more than 10% right.... anything that B calls an all in with here has player D pretty much dead!

So let us have a look at the 'worst case scenario' for player A - after calling and losing to B on the flop.

Player A - 2000 - $ev = $28.62
Player B - 6500 - $ev = $40.78
Player C - 2500 - $ev = $30.61

So now the final numbers part: From the perspective of player A - pushing top pair or better on the flop.

95% of the time he wins/loses 75/25% - average equity gain = +$1.77c
5% of the time player B gets a monster and wins - equity loss = -$7.59

(95*1.77)-(5*7.59)/100 = $1.3

Its clear then that isolating with that top pair is a positive expectation move - even in an 'obvious' check-through scenario. In addition the benefit of having 6000 vs 2500 + 2500 in the money is huge from a strategic standpoint...

To conclude part #1 then: suprised myself here - thought I'd be using this example as a 'standard' to compare some other situations and it turns out differently than expected. Damn it I have checked through overpairs in this same spot several times costing myself money!!!

One other factor to consider is ths critisism you'll get from the other players in the chat for such a move... they will not have thought through the maths!! If you care about that more than making money please leave your Stars or Titan Poker screen name in the comment box - I would very much like to play at your tables!!!

GL at the tables.

Mark

1 comment:

Anonymous said...

yeah, but most of the time you will not have TP (or better) on the flop.
So most of the time, you'll be checking it down because you want the shorty busted (unless you want to keep him in so you can bully).

Point is, the only time a push is worthwhile is if your hand is better than the shorties', but worse than the medium stack (eg TP worse kicker)... i.e. you get the medium stacker to fold with the best hand, and you win against the shorty.
This situation would happen rarely... and is not worth analysing too much.