PezRez on the 17th August 2011
**WARNING: THIS POST CONTAINS MATHS**
A situation came up for me recently that left me with a ridiculous choice to make. It’s the bubble and on the button with 4bb I‘ve got to shove any two. I found T2o and shoved it - or meant to, but I didn’t quite slide the bar all the way up and so raised to 700, leaving myself with 100 behind.
Then something incredible happened: the middle stack shoved and then the big stack shoved over the top! I watch in glee (“They’ve got it all in!”) - and then realise the action is back on me. Oh yeah, I opened this hand. And I have 100 left. All the chips are in the middle, and I have 23-1 pot odds to call all-in here with T2o. It seems like a no-brainer, but I just clicked fold and coasted into the money. Here’s the HH:
PokerStars Game $93.25+$6.75 USD Hold'em No Limit - Level VI (100/200)
Seat 3: MidStack (1515 in chips)
Seat 5: BigStack (6685 in chips)
Seat 6: Hero (800 in chips)
MidStack: posts small blind 100
BigStack: posts big blind 200
*** HOLE CARDS ***
Dealt to Hero [Tc 2d]
Hero: raises 500 to 700
MidStack: raises 815 to 1515 and is all-in
BigStack: calls 1315
*** FLOP *** [Jd 4h 5h]
*** TURN *** [Jd 4h 5h] [Qs]
*** RIVER *** [Jd 4h 5h Qs] [Kd]
*** SHOW DOWN ***
MidStack: shows [Ad 5c] (a pair of Fives)
BigStack: shows [7s 7d] (a pair of Sevens)
BigStack collected 3730 from pot
MidStack finished the tournament in 3rd place
*** SUMMARY ***
Total pot 3730 | Rake 0
Board [Jd 4h 5h Qs Kd]
Seat 3: MidStack (small blind) showed [Ad 5c] and lost with a pair of Fives
Seat 5: BigStack (big blind) showed [7s 7d] and won (3730) with a pair of Sevens
Seat 6: Hero (button) folded before Flop
Now I revisited this hand at the end of the day and really wanted to know whether you should call or whether this ridiculous fold with 23-1 pot odds, leaving me with half a big blind, could actually be correct!
Unfortunately, ICM calculators like SNG Wiz aren’t capable of dealing with situations where you have already acted in the hand, such as here. In this situation, we need to do the ICM calculations longhand. Here it is also complicated some more by the fact we have two possibilities to consider: our probability of winning if we call (which we’ll label p) and the chance the Midstack busts if we fold and therefore we sneak into the cash (let’s call this one x).
For those of you who are not too bothered about the maths of poker, skip to the ending. To the rest that remain, let’s have a go at this problem.
So first off we are going to need an ICM calculator, such as can be found at holdemresources.net. Thankfully this does the really hard stuff, so we are left to concentrate on only the moderately hard. You stick the stack sizes and payouts into the calculator and it tells you each player’s equity, according to ICM. We can use this to figure out what our equity will be in various outcomes, and then construct an equation to compare.
If we fold, we are going to have 100 left after the hand. If the big stack wins, we’re in the cash! But if the midstack wins, with 100 left we are going to be f♠♦♥♣d.
The ICM calculator tells us that our equities in those two situations are:
EQ[we fold, bubble pops] = 35.33%
EQ[we fold and midstack wins] = 1.55%
1.55%. That’s what f♠♦♥♣d looks like.
Here we need to make an assumption to simplify things. We are going to assume that we know what their hands are in this situation and if we win the main pot (most likely by spiking a Ten) then the Big Stack will win the side pot, as it will mean that the Mid Stack failed to spike his Ace and so came third in the hand to my pair of Tens and the Big Stack’s 77. This will nearly always be the case, unless we get some funky board like A55 or TT55x or maybe AT2. These things are gonna happen, but very infrequently (like maybe 5% of the time), so let’s discount them. As for how this will bias our calculations, it will slightly overestimate our equity for calling and so we’ll end up with a slightly lower winning percentage we need to make the call.
So either we triple up and end up heads-up with Big Stack or we bust and get nothing. The ICM calculator gives our equity in the first case as:
EQ[triple up, make the cash] = 44.00%
And it should be obvious that if we bust, our equity is zero.
To analyse whether calling is better than folding, we need to set the equities of the two against each other in an equation and then solve it. The probability we calculate will be the winning chance above which calling will be better than folding. But since we have both p (our chance of winning) and x (Big Stacks chance of winning when we fold) in this case, we cannot “solve” the equation. We will have to introduce some likely values for one to look at what that would make the other.
So setting EQ[FOLD] = EQ[Call] will mean the following equation:
EQ[FOLD + CASH]*x + EQ[FOLD + F♠♦♥♣D]*(1 - x) = EQ[CALL + WIN]*p + EQ[CALL + LOSE]* (1 - p)
Putting our numbers in:
35.33*x + 1.55*(1 - x) = 44*p + 0
33.78x + 1.55 = 44p
Now we have our equation, let’s put in a couple of values. First of all, the Big Stack here is likely to have a tighter range than the Mid Stack, and he’s a much less fishy player, so let’s assume he is a 2-1 favourite when we fold (that is, x = 0.67). In that case:
35.33*0.67 + 1.55 = 44p
25.22 = 44p
p = 0.5732 or a 57.32% chance of winning
Well look at that! If the Big Stack is a good favourite here, you need a 57% winning chance to call - and I’m pretty sure that’s impossible with T2o in a three-way pot. It’s not just a fold - it’s a huge fold. And remember, because of the bias introduced by our assumption, this number is an underestimate.
What about if the Big Stack is not such an extreme favourite? Let’s see what happens if the two of them are all-in for a coin flip (x = 0.50):
35.33*0.50 + 1.55 = 44p
19.22 = 44p
p = 0.4368 or a 43.68% chance of winning
So whilst that has made a big difference, again you’re never gonna be winning 44% of the time with T2o in a three-way all-in. Remember, with a random hand versus two other random hands you are 33%. Here you have a decidedly below-average hand against two clearly well-above-average hands. Still a huge fold.
In fact, against a range of the top 20%of hands for the Mid Stack and the top 15% for the Big Stack, your T2o has a 19.5% winning chance. With these ranges, when we fold the Big Stack has 54.9% equity in the pot. So looking for x = 0.549 and p = 0.195...
EQ[FOLD] = 20.10%
EQ[CALL] = 8.58%
Calling here would cost me 11.52% equity, equivalent in this $100 buy-in game to $64.45!
RESUME HERE IF YOU SKIPPED THE MATHS
These calculations have proved that in this situation it is in fact correct to fold the hand, despite your 23-1 pot odds and half a big blind remaining. Not only that, but it would be a grave error to call here, costing me in fact almost two-thirds of my buy-in in equity if I call. SNGs, whilst being such a repetitive and formulaic format, never fail to amaze me sometimes. Never would I have thought that calling with 23-1 pot odds could possibly cost you $65 in a $100 buy-in game. If you take anything from this post, it should be that in SNGs, ICM is King.