Small Stakes No-Limit Hold'em

The discussion on page 162 about the equity of the all-in 4 bet seems either incorrect or misleading.

That discussion begins by assuming you are on the button and an opponent 3-bets 18% of hands, but calls an all-in 4-bet with AA-TT, AK. I will assume the opponent is in the big blind, although it is not stated if he is in the SB or BB, and the blinds are 1/2.

You have AQ with $200 stacks and (open) raise to $7 on the button. SB folds, and BB raises to $22.

The text states “If you simply push all-in, your line yields about $5.20 in average equity for you.” But in this situation, the player is interested in whether to push or fold, presumably. He has already put the $7.00 in to the pot, and that decision is not going to change.

So the more natural to ask is, what is the equity of pushing? The equity of pushing can be computed in poker stove as follows.

First, I assume by “AQ” you mean “AQo” here, since the text does not state AQs. Say it is AcQd. It is important to remove these two cards from the distribution of the opponent; that is, the opponent is less likely to have AA now. This means that the opponent is only pushing 36 hands: 3 AA, 3QQ, 6 KK, 6 JJ, and 12 AK. There are 1326 hands, so this is a fraction of 36/1326=.027 that he is going all-in with. Since he 3-bet with .18 of the hands, the probability he will call the all-in is .027/.18=.15.

Anyway, carrying out this computation in pokerstove, your equity, once called, is about .306 times the pot, which is $401 (the stacks plus the small blind).

When the opponent folds, you make 22+1+7 or $30.

When the opponent calls, you make .306*401-193 = -$70.294
Since the opponent calls .15 of the time and folds .85 of the time, the total equity of pushing all-in 4-bit is;

equity due to fold + equity due to call =

.85*$30 + .15* (-$70.294) =  $14.956

Thus, every time you go all-in, you are making on average about $14.956 (less rake, of course). It’s true that the equity of the entire line, that is that includes the inital $7.00 bet, would be exactly $7.00 less than this, or about $8.00 total. But this is not relevant to the question of whether to shove or not. That still leaves an unexplained difference of $2.80 in our computations – perhaps I made a typo somewhere.

The main point here is that I think it is critical to carefully distinguish between the equity of the push and the equity of the line.