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Originally Posted by Cobra_1878
This is going to sound strange but I actually like Maths; my mental arithmetic (and a lot of other Maths related topics) is really, really good. It's just when it comes to Algebra and sorting out sums/equations that involve brackets I just can't seem to wrap my head around it. I would be more than happy to give it another go if you want to show me MMM?
I am dedicated because I love playing poker. I realize that equity is ridiculously important in poker it's just something that I struggle with.
OK, so MMM might be able to explain it even better, but here's my stab at doing it without any unnecessary brackets:
If villain never folds, so he either calls and we win, or he calls and we lose, then say we're 100bb deep each, and no-one else is involved in the pot (let's say we're the SB and BB) so we don't have to worry about any other bets that have gone in, or the blinds, so the pot is always 200bb.
If we win, we win 100bb
If we lose, we lose 100bb
If our equity is 50%, then 50% of the time we win, and 50% of the time we lose, so over time, we break even. Another way to look at this is to say that each time a 200bb pot takes place, we "win" 50% of it (100bb), which also says we break even - we shoved in 100bb, and we "won" back 100bb of expected value (even if on this particular occasion we either got 0 back, or 200bb).
Say our equity is 25% instead, now we win 100bb one quarter of the time, and three quarters of the time we lose 100bb. So in four repeats of this scenario we'll lose a net 200bb, so on each of the occasions we'll lose an average of 50bb. We shove in 100bb, so does villain, there is a 200bb pot, and our equity share (25%) of that pot is 50bb, which means we lost 50bb (because we shoved in 100bb, to get back 50bb).
So let's take it a stage further. We win the proportion of the time that we have the equity to match (half the time if we have 50% equity, a quarter of the time if we have 25% equity, 3/4 of the time if we have 75% equity etc. etc.). So if we use E for equity, then we lose 1-E of the time (ie. if E=0.25 meaning we have 25% equity and win a quarter of the time, then we lose 1-E=0.75 which is 75% of the time).
If we're analysing shoving, then when we win, we win the current pot plus the remaining effective stack, and when we lose, we lose the remaining stack. That is to say, if there's money already in the pot, then it's no-ones money any more - it's already in the middle, so if we shove, get called and lose, all we lost by doing that was whatever we shoved - the rest was already lost. If we shove, get called and win, then what we actually win is whatever was in the pot when we shoved and also what got called (the effective stack at the time of shoving).
To put that in algebra form, when we win, we win P+S (pot+stack), and when we lose we lose S (stack).
So, if villain ALWAYS calls, the total EV of shoving is:
win% * P+S - lose% * S
Ie. some percentage of the time we win P+S, and some percentage of the time we lose S. win% is just our equity. lose% is every time we don't win, ie. 1-E. So we can rewrite that as:
E*(P+S) - (1-E)*S
That's the total EV of shoving and always getting called. It's what we win (or lose) on average each time we shove and are called.
But the other way we can win the pot is that we shove and he folds. So if that happens, we just win P (the pot at the time we shove). So the bit of the EV that comes from him folding is just:
%fold * P
But he doesn't always fold. So some percentage of the time he folds, and the rest he calls. We'll call the percentage of the time he folds FE for fold-equity, so eg. FE=0.3 says that he folds 30% of the time, so then he calls (1-FE)=0.7 70% of the time. He either calls or folds - they must add up to 1, so he calls with frequency 1-FE.
So we stand to win P * FE because he folds (the pot, times the percentage of the time he folds), the rest of the time (1-FE) we'll get called and the bit of the equation further up this post applies.
So
FE * P
+
(1-FE) * (E*(P+S) - (1-E)*S)
Or, to reword that in english:
The % of the time he folds (FE) we win the current (small) pot (P)
+
The rest of the time he calls (1-FE) and we win/lose according to our equity share in the big pot (E*(P+S) - (1-E)*S)
I dunno, it's hard to explain it entirely without brackets. Hopefully that might make it a bit clearer.
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