Push EV calculations for BvB situations

[Garth Kay]

Hi All,

Welcome to my first "advanced" article; I thought I would start off slowly with some simple calculations and formula. In this thread we discuss PUSH EXPECTED VALUE and we need to decide whether a push from the SB will result in +EV for us.

If you are unsure what EV is then I will briefly explain that EV is our value (whether it is positive or negative) over time if we made the same play. EV will become more clear as we progress through the calculation and example, but if you're still unsure about this then please ask and I will create another article on EV (EV is the basis of the majority of all Math based poker decisions, the ultimate goal is to always make +EV decisions as a poker player).

So back to PUSH(shove) EV. Whilst we can work out our shove EV for a multiway pot, to keep things simpler for the short term we will use the scenario that action is folded to us in the SB and we are contemplating shoving in this spot.

So to work out our shove EV we need to know several things:

*X = Our stack size (Always try to do this in BB's)
*C = Percentage of the time we will be called
*The range of the villain and therefore
*W = Percentage of the time we win when we are called (POKERSTOVE is required for this; it is the must have tool for the serious poker player, even if you don't have HEM or PT you should have Pokerstove)

So to calculate our EV we use this formula:
[(1 - C)*1.5] +[W*C*(X+0.5)] - [C*(1-W)*(X-0.5)]

The 1.5 above refers to the amount of big blinds we will win when we successfully steal.
X + 0.5 = the amount we will win when we are called and win
X - 0.5 = the amount we lose when we are called and lose.

So basically the formula above equals the following:
Amount won when BB folds + Amount won when BB calls and loses - Amount lost when BB calls and wins

And this formula can be stripped down to the following:

SHOVE EV = (1.5 - C) + [C*X*(2*W-1]
X= stack size
C = % of time you are called
W = % of time you win when you are called

(NOTE: this is for only SB vs BB situations and does not take into account other positions and it is believed this is the opening action)

EXAMPLE

Action is folded to us in the SB we are holding 78 of spades and we know our opponent will call off with a range of 22+, A8s+, A5o+, KTs+, KJo+ and QJ.

Poker stove then says:

Text results appended to pokerstove.txt

426,363,696 games 0.633 secs 673,560,341 games/sec

equity win tie pots won pots tied
Hand 0: 38.383% 37.96% 00.43% 161836831 1815322.00 { 8s7s }
Hand 1: 61.617% 61.19% 00.43% 260896221 1815322.00 { 22+, A8s+, KTs+, QJs, A5o+, KJo+, QJo }

That we have 38.383% equity in the hand (we will win 38.383% of the time).
Therefore W=0.38383

We know the hand range assigned to the BB is approx 20% of hands (this can also be found in Pokerstove) but to calculate it without pokerstove we calculate the combination of hands in the ranges using another formula (Won't put it here as it will confuse too much).

So therefore C=0.198

Know my stack size is 12BBs; so X=12

So therefore SHOVE EV = (1.5 - C) + [C*X*(2W-1)]
= (1.5 - 0.198) + [0.198*12*(0.76766 -1)
= (1.302) + [2.376*-0.23234]
= 0.7499

So we can decide that this push is +EV as we will win 0.75BB's every time we are in this situation.


HOMEWORK

Action folds to you (Hero in the SB) and you are facing a villain with the following stats (24/18/3.6 - VPIP/PFR/3b).

IMPORTANT DATA:
You have 16 BB's and holding J8 Diamonds, the BB has 23BB's

INTUITIVE THINKING
Is required in this situation, you will need to assign the calling ranges of the villain in this scenario.

QUESTION
Do you shove?

ANSWERS and WHAT YOU WIN
Please PM me your answers, do not POST answers in here as I will delete them. The person who shows all the correct working and resolves the best hand ranges for villain will be awarded a nice present at the next Grand Final!

If there is a problem with formatting or more answers required let me know.

[Garth Kay]

One answer in!

Anyone else willing to have a go or am I wasting my time?

[bennymacca]

One answer in!

Anyone else willing to have a go or am I wasting my time?

ill definitely have a go, haven't yet. will definitely do tonight or tomorrow night

[trishan]

One answer in!

Anyone else willing to have a go or am I wasting my time?

Yep, I'll be sending an answer through when I get back home later.

[Swanky]

I hate maths, with a passion. But this is something I really want to have a crack at learning.

I'll keep you posted.

[John Miller]

Will have a crack later on today...if i have time, have a tourny to run tonight so might be tomorrow morning.

[Origami]

final draft to be submitted within minutes..

[trishan]

We can take this equation a step further and try to figure out what is the minimum equity we require to shove given our stack size and the calling % of the BB. I will try to add as many steps as possible to the algebra.

Shove EV = (1.5 - C) + CX(2W-1)

Shove EV must be greater than or equal to zero to make a call + EV.

(1.5 - C) + CX(2W-1) ≥ 0
CX(2W-1) ≥ C - 1.5
2W-1 ≥ (C-1.5)/CX
2W ≥ (C-1.5)/CX +1
2W ≥ (C-1.5)/CX + CX/CX (As 1=CX/CX)
2W ≥ (C- 1.5 + CX)/CX
2W ≥ [(X+1)C -1.5]/CX (Grouping C terms)
W ≥ {[(X+1)C -1.5]/CX} * 1/2
W ≥ [(X+1)C -1.5]/2CX

Let's say our stack is 15BB, then we can graph the relationship between W and C:



The X-intercept represents the value of C where we can shove any two cards. It looks like it is about 8%. So if there is someone so tight that they would only call a 15BB jam 8% of the time in the BB (88+,ATs+,KTs+,QJs,AJo+), it is always +EV to jam.

You can also see that if C is between 10% and 30%, the equity requirement to jam is not very high at all.

[trishan]

You can also see graphically how different stack sizes affects the minimum equity required for a shove to be +EV:



15BB
20BB
25BB
50BB
100BB

You can see that as your stack size increase you require a much greater minimum equity to profitably shove in the SB.

[trishan]

So how did we all go with the question?