Blind vs. Blind

Here's a weird one I discovered while noodling around with a new equity calculator I downloaded recently*. The result I get is strange enough that I'm actually not fully convinced it's true. If you see a flaw in my math or logic, please email or leave a comment:

Scenario: You're in the Big Blind and the action folds all the way 'round to the Small Blind. He's a regular ABC TAg player, and he opens to 3x.

Question: From a theoretical math point of view, how often should you call?

Short Answer: A lot.

Longer Answer: A helluva lot.

Here's Why: For argument's sake, let's assume that the SB is not opening crazy wide. Call it all Broadways, all Pocket Pairs, and all Aces. This is just under 28% of hands (366/1326), and this is what it looks like in a card matrix:

When the villain opens to 3x, there is 4x in the pot (his 3x open, plus your 1x big blind). It's going to cost you 2x more to call, so you're getting 2:1 direct pot odds. This means you need 33% pot equity to breakeven if you were to call and the hand runs out hot-and-cold to the river (1/(2+1)). If your pot equity is higher than this, it's profitable to call; if it's lower, than it's not profitable.

In terms of range, what does 33% equity look like against villain's opening range? My assumption is that I'm re-raising the top 15% of my range (all but the worst Broadways, and 77+). Everything else is fair game to just call with. So I started opening up a range of hands against Villain's range in Equilab...

... and I was pretty shocked when I ran the numbers. I basically kept adding cards, trying to drive my equity down to 33%. Turns out I couldn't. The closest I could get was 35%, and that is with this range:

Yep, this means when I'm not 3betting the premium hands, I should be calling. Period. Yep, all 100% of my range. Yep, seriously. And it gets even more pronounced with opening up Villain's range wider. Seriously. For instance, if Villain opens with, say, 35% of his hands, I automatically get about 38% equity with any two cards. Change his range to a LAggy 50% range, and my equity jumps to 42% with any tow random cards. And so on...

Now, this is the theoretical math solution. The reality is that I would be psychologically hard pressed to call with total junk hands like K3o and the like, as they're actually pretty hard to play post-flop, even in position. (Remember: this exercise assumes all post-flop action is moot, and the hands run hot-and-cold to the river as-is.) But still... this is pretty amazing.

Another wrong message to take from this example is to start defending your blinds lightly against button openers. You'll be OOP throughout the hand, which will have a serious impact on your ability to play postflop poker. You'll need more equity than just 33% to make defending against a button steal.

Anyway, I found this fairly interesting... but like I said, I'm not convinced I didn't do something dumb along the way in my analysis. Any and all comments are welcome!

All-in for now...

-Bug

*PokerStove does not seem to be available for download anymore. That's the bad news. The good news is there's an equally good (and still free) program that does what PS did (actually the new software does more). The program is called Equilab, and it's created/provided by Pokerstrategy.com. Click here for the link to download it.