2.1. Calculation

Hero has the stack S1 and plays against Villain with stack S2, with S being the smaller of the two stacks. Hero pays the SB, Villain the BB, both receive their cards and Hero ponders about either pushing or folding.

If Hero folds, his Expected Value, which we will call EV, is obviously

(0) EV =-SB.

(In comparison to his stack prior to paying the blinds he exactly loses the SB he had to pay.)

If Hero decides to push, the following might happen:

(1) Villain folds
(2) Villain calls and loses
(3) Villain calls and wins

How do we calculate the EV now for Hero's push?

It is made up of the three “parts” EV1, EV2, EV3, according to the three possible outcomes (1) (2) and (3) combined:

(1) Villain has a certain calling range (more about the observation of different calling ranges later on). All hands outside of this range will be folded by Villain against our push. The probability for a fold, so for a hand outside of his calling range, shall be F.

Therefore: EV1= F*BB

(Hero receives the opponent's BB and his own SB. His initial stack S – prior to paying the SB – thus increases by BB)

(2) Villain has with the probability C=1-F (C stands for Call) a hand of his calling range. Against this we have a certain win probability W. If Hero has e.g. 88 and faces TT, we can roughly calculate: W=19%

EV2=C*W*S

(When Hero gets called and wins, he receives S chips from Villain.)

(3) Villain has against Hero a win probability of (1-W). Thus the following equation applies:

EV3=C*(1-W)*(-S)

(When Hero gets called and loses, Villain receives S chips from Hero.)

For the total EV of the push holds true: EVt = EV1+EV2+EV3

We can put this on a level with the EV of the Alternative Fold EV and receive:

(*) -SB = F*BB + C*W*S + C*(1-W)*(-S)

Now we can solve via the value S/BB and receive:

(**) S/BB = (F+0.5) / [C*(1-2*W)]

For a given calling range we can calculate C and W relatively easy:

• C is the amount of hands from the Villain's calling range, divided by the amount of all possible hands. The later is 50*49/2 = 1225, as Hero already has two of the 52 cards.
• W can be calculated with a program like Poker Calculator or PokerStrategy.com Equilator, by including every hand of Villain's calling range in accordance to Hero's hand. Thus W always depends on Hero's hand of course.

If we included these values in (**) (for F we insert 1-C), we would receive a ratio of S to BB. Thus a value how big S, expressed in BB, can be at the most for the push not to become -EV. Or put another way: When the actual stack is smaller than this value, then the push is +EV.

Example:

Hero holds ATo. He assumes that Villain's calling range is 66+, AT+ and A8s+.

First we figure out the amount of hands in Villain's calling range:

As Hero has an A and a T, there are only three different combinations for AA and TT respectively, for any other pocket pairs 6 different ones, so overall 7*6+2*3 = 48 pocket pairs.

The aces AK, AQ and AJ, for which it shouldn't matter if they're suited or not, give 12 different combinations, as the A in Hero's hand already eliminates four of the general 16 possible combinations. For AT, the T in Hero's hand thwarts another three combinations. A9s and A8s finally add another three combinations respectively.

Overall the calling range has 7*6 + 2*3 + 3*12 + 9 + 2*3 = 99 hands. From this we get

F = 1 - (99/1225) = 91.92%

For W we have (with help from PokerStrategy.com Equilator or similar):

W = 35.5%

Included into the equation (**) it results into:

S/BB = 60.6

If the opponent's stack or your own so the actual stack, is smaller or equal to 60.6 BB, then Hero is able to push +EV. The results depend on your own hand as well as on the opponent's calling range.

Equation (**) goes towards infinity, if 2*W goes towards 1 or F goes towards 1.
This makes sense since: If we have a winning probability W of at least 50% against Villain's calling range, we are able to push with any big stack. This is also true if he folds with a probability F=1, so every time.

The situation W >=50% would be the case if Villain calls with every hand and Hero has an above-average hand (everything above Q5) or when Hero has AA. The situation F=1 happens for example when the opponent is no longer at the table. Unfortunately, it'll never be that easy.