Imagine an opponent who offers you a very simple hold 'em game. You are the small blind, and you get to go all-in or fold. If you go all-in, your opponent will call if he has kings or aces, and will fold if he has any other cards. You are sitting with 200 big blinds, and you move in preflop. The question is, would you rather have 76s or AKo?
It seems pretty simple to calculate your expected value in this situation. AA and KK represent 12/1326 or .9% of possible hands, so if you push, 99.1% of the time, you will win 1 BB, and .9% of the time, you will lose (200BBs - (your hand's equity vs. AA/KK*400BBs).
Programs like pokerstove can judge your equity against a range of aces or kings. It turns out that 76s has .225 equity against aces and kings and AKo has .185 equity. When your opponent calls you, you lose 126 BBs with AKo and 110 BBs with 76s. Therefore, it would seem that you would rather push with 76s than with AKo; in fact, it appears to be -EV to push with AKo, while it is barely +EV to push with 76s. (-.143 BBs vs. +.001 BBs).
However, this is not the case. This simple analysis neglects card removal effects. When you are holding a card in your hand, your opponent can't have that card in his hand. Ordinarily, there are 6 ways your opponent can be holding a pair of aces, and 6 ways your opponent can be holding a pair of kings. However, if you're holding an ace and a king in your hand, there are only 3 ways that your opponent can have AA and 3 ways for him to have KK. With this extra information, his chances of holding AA and KK are almost halved. (Instead of a 12/1326 chance, he now has a 6/1224, or .49% chance). If you instead have a 7 and a 6 in your hand, the chances of your opponent holding AA and KK are.98%, since he has the same 12 combinations of aces or kings, but you can rule out several combinations of hands with 7s or 6s.
When all the math is done, your EV is much higher pushing AKo than pushing the 76s. Accounting for card removal, pushing with 76s is now worth -.09 BB, and pushing with AKo is worth +.38 BBs.
It's common to ignore these sorts of card removal effects for two reasons. First, they are rarely this pronounced. They only have a strong impact when an opponent's possible range of hands is very narrow, and when many of his hands rely on the presence of a key card or cards (the ace and the king). The second reason is that many poker analysis tools, including pokerstove, already handle card removal effects in equity calculations. If you match up AJo vs. a range of AA-KK, pokerstove will account for the fact that you are more likely to be up against KK than AA...
Keeping that in mind, I'm going to look at two situations where card removal seems to have a measurable effect: first, dealing with a preflop raise with a moderately tight player defending a blind, and second, putting in a 4-bet against a loose reraiser.
Making an opening raise, with or without a high card
Our first scenario: you are in the small blind. Your opponent in the big blind is moderately tight, although he plays a lot of high card hands. He defends his blind with 22+ A5+ K9+ Q9+ J9+ T9 A2s+ K4s+ Q7s+ J7s+ T7s+ 96s+ 86s+ 75s+ 64s+ 54s and folds other hands. So, in a vacuum, he plays 36%, 478 combinations of his hands, and folds the rest. Furthermore, he reraises with 88+ AJ+ ATs KQ.
Let's say you are contemplating a preflop raise with one of two garbage hands, 43o and K2o. What are the implications from a card-removal standpoint?
If you are holding 43o, the opponent plays on 467/1224 or 38.2% of the time. He reraises110/1224, 9% of the time.
With K2o, the opponent plays on 446/1224 or 36.4% of the time. He reraises 99/1224, 8.1% of the time.
If you open with a pot-sized raise, break even postflop when flatcalled, and fold your hand when reraised preflop, then in this scenario K2o is worth +.393 bb/hand, and 43o is worth +.348 bb/hand, for a net difference of .045 BB/hand, or 4.5 bb/100. (The effect is more pronounced with an A2o, which under the same assumptions is worth a 6.8 bb/100 premium over 43o.)
It's also useful to note that the more aggressive your opponent is preflop, the more value the high-card blocker(s) in your hand are worth. If you are playing a demented opponent, who, instead of flatcalling, reraises every time preflop with the same 38% range listed above, then 43o is worth -.528 bb/hand (you should fold your small blind), K2o is worth -.464, and A2o is worth -.41; the spread is now almost 12 bb/100 between the value of the hands.
Pushing over a reraise, with or without an ace
In this scenario, you make an initial preflop raise, face a 3bet, and are considering coming over the top with a weak hand, hoping your opponent will fold. I'll use the setup from last month's 2+2 magazine article by Bryce Paradis and Dusty Schmidt. You make it 3.5 BBs to go, your opponent makes it 12 BBs to go, there is 1 BB of dead money in the pot, and you both started with 100 BBs behind.
Let's say you're facing an aggressive and slightly tricky opponent who reraises with 16.1% of his hands, including many semibluff hands of his own: small pairs and suited connectors. His range for making a reraise is 22+ AJ+ KQ+ KTs+ Q9s+ J9s+ T8s+ 97s+ 86s+ 75s+ 64s+ 54s. Let's give our opponent a range to call reraises of AQ-AK, 88-AA. He's calling reraises with the top 5.6% of hands, 74 combinations, 34.6% of the 214 combinations he raised with.
You are considering coming over the top of his raise by moving all-in, with either 22 or A2s. 22 and A2s have similar equity vs. our opponent's range of hands: A2s has .307, and 22 has .327 equity if called. According to a strict equity-based analysis, 22 should be the better hand to push, and, factoring in dead money, would be worth .14 BB, while A2s would be worth -1.25 BB, However the card removal effect of the ace is significant.
Including card removal effects, holding 22, your opponent calls with 74/209 combinations, or 35.4% of the hands that he reraised with. Holding A2s, he calls with 63/196 combinations, or 32.3% of the hands that he reraised with.
Now, the value of the play with 22 is -.23 BB, and with A2s is +.04 BB. So the card removal effect of the ace is worth 1.29 BB, enough to turn a push into a slightly profitable move for the A2s, while the card removal effect for the 22 has a slight impact in the other direction, -.37 BB, enough to make it unprofitable to push.
The lesson to take from all this math is that that high card hands, particularly if they contain an ace, have a little extra value preflop beyond what is normally considered, since they make it more likely that your opponent will not have a playable hand, while hands that contain two low cards have a little less value preflop than what is normally considered, since they make it more likely that he will have a playable hand. While the difference is small in many spots, the effect of an ace in particular can be substantial as the opponent's range of hands to call with becomes narrower.