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How do I calculate my expected value of shoving, including Fold Equity, in heads up play?

I know several factors are involved

  • Pot Size
  • My Stack
  • His Stack
  • My chance of winning if he calls
  • His chance of folding when I bet
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Expected value of what kind of bet? All-In preflop, regular bet/ all in on flop, or on later streets? – MartinK13 Jan 10 '12 at 22:53
Sorry, lost that on an edit. Fixed – Frankston Ralphington III Jan 10 '12 at 22:57

2 Answers

up vote 9 down vote

This will be pretty messy if I don't define some variables, so here goes:

  • P$ = Current size of the pot
  • S$ = Minimum of your stack vs your opponent's stack
  • F% = Chance of your opponent folding to your shove (this should be between 0 and 1; divide percentages by 100 to get corresponding value)
  • W% = Chance of you winning when called (this should be between 0 and 1; divide percentages by 100 to get corresponding value)

When he calls and you win, you earn:

P$ + S$

When he calls and you lose, you lose:

S$

So, if he always called, your EV would be:

EV = (W% * (P$ + S$)) - ((1 - W%) * S$)

But he doesn't always call. When he doesn't call, you win:

P$

So we can add that into the above:

EV = (F% * P$) + (1 - F%) * ((W% * (P$ + S$)) - ((1 - W%) * S$))

I used this formula for the basis of a Expected Value Calculator web-app at Grinderschool, which calculates EV, Required Fold %, and Required Win % given any/all of the variables represented in this equation.

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1  
I intend to make a chart out of this formula, which should make the concept easier to visualize and, thus, memorize. I shall edit it into the answer if I get around to it before somebody else. – Frankston Ralphington III Jan 11 '12 at 3:05
I like what you've done here. Note that the equation can become even more precise by incorporating the possibility that the pot will be split. – Chris Dec 22 '12 at 3:14
up vote 1 down vote

The EV is (% he folds to All in * Current pot size) + (% of times opponent calls * % you will win * Total size of pot) - (% of times opponent calls * % you will lose * Amount that you bet/shove).

On the left of the "+" sign are the times without a showdown. On the right are the times with a showdown. The times you win or lose can be calculated either against his explicit hand, or against what you assume his range is.

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There are two problems with this: First, when you lose, you did not lose the entire pot size; you only lost the amount that you bet. Second, this does not account for the fact that he only sometimes fails to fold (e.g. you must devalue the amounts you win and lose when he calls by the frequency that he folds). Note that the existing answer properly accounts for both of those factors. – Jeffrey Blake♦ Jan 12 '12 at 7:33
@JeffreyBlake Just edited. I noticed the mistakes I made, good catch. Thanks :). I'm paranoid now, this is right? – Toby Booth♦ Jan 12 '12 at 14:04
Toby, yes, that is right. It also matches my equation above, save that it converts the (1-x%) values into new variables. That's unnecessary, since the two are entirely related. – Jeffrey Blake♦ Jan 12 '12 at 16:00

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