All-In EV and The Bunching Effect

Let's just say (as an example) that in blind battles, or button vs blind battles, in hands where it gets to the button or the blinds unopened, you'll pretty much always get it in with AT+, maybe A7s+. And let's say your buddy will go with only his best Aces but will get it in with any pocket pair. You are going to run better, over the long run, in all-in EV than your buddy.

Because, on average, there will have been more opens from earlier positions by people with aces in their hands than any other card. So in this situation, where no one has opened yet, you'll have more than your share of aces left in the deck, and he'll have less than his share of cards to make sets.

Let's say AJo v 77, AIPF. AJo has 45.352% equity (a bit lower than people tend to think -- if you switch it to suited it gets up to about 48%).

Now, if we remove one of the two remaining sevens in the deck, AJo climbs all the way up to 50.04%. That's a pretty big difference, considering we're not really thinking about avoiding a set when those cards get turned over.

Now, if it's a fullring game and seven players folded before the action started, and none of them had an Ace? With 14 non-Ace cards removed, the AJ0 gets a slightly bigger bump, up about five points to 50.685%. If we make one of those removed cards a seven, it surges all the way to a 57.166% favorite.

So, just based on card removal, the hand can pick up about 12% in equity. Meanwhile, all-in EV continues to calculate your equity in the pot based on all cards being available. So, over not much time at all, the person with AJo will begin to seem luckier than the person with 77, as he drags a higher percentage of pots than the (thought to be infallible) EV calculator predicts he will.

Of course this has all been extreme. I didn't take out any J's, either, which is just basically wrong. And there will sometimes be Aces in the muck, just like often there won't be any 7's there. So the actual impact of card removal is obviously far less than 12%. But it's definitely far greater than 0%, too. Which means a player's style can affect how well they run in all-in EV (as calculated by HEM, or similar programs).

I have no idea if in reality there would be many instances where a player's style deviates from averages enough to start having card removal really affect how he runs in a meaningful way, but I now know that, theoretically at least, it's possible.