A reconsideration

Last week I described a heartbreaker of a hand against Spewtard. The story, in short, is that I had Qd-Qh, and I knew that he had exactly Ad-Jd. I raised (to $16, as I recall). He reraised to $40. I raised another $100 on top of that. He shoved for a total of what proved to be $435. I had him covered and called. The flop and turn were safe, but he caught an ace on the river to win.

In the comments, somebody asked about the alternative strategy I could have taken: Just call the reraise to $40, then decide on the flop whether to commit. I had, in fact, briefly considered that option as the hand played out, but made the instantaneous decision that I knew I was roughly a 2:1 favorite, and I should get in as much money as possible with those odds.

However, I've been debating this on and off ever since, and I wondered if I could work out the expected value of each strategy. I think I have done it, to a reasonable approximation.


Strategy 1

This is what I actually did. I was a 68.3% to 31.7% favorite to win. So if we did the hand 1000 times, I would win $435 683 times for a total profit of $297,105. I would lose $435 317 times, for a total loss of $137,895. Net gain would be $159,210. So my expected value (EV) for this strategy is approximately +$159 for each time we play the hand this way.


Strategy 2

The particular Spewtard was so bad that he very well might have been willing to shove on the flop no matter what cards came, and whether I checked to him, bet small, bet big, or shoved myself. I'm using that assumption here.

Let's suppose that I decide to shove (anticipating a call) on any flop where I'm ahead, and check-fold on any flop where I'm behind. How would that work out mathematically?

Here are the types of flop where I'm behind, and the number of possible flops that would fit each category, given the four known hole cards:

3 aces: 1

2 aces: 135

1 ace: 5440

3 jacks: 1

2 jacks (no ace): 126

K-Q-T: 32

3 diamonds: 56

Note that I don't have to account separately for two-pair combinations, because any flop that pairs both of his hole cards necessarily contains an ace, which alone is enough to beat me and cause me to fold.

That's a total of 6291 flops in which I'm behind and will fold, thus losing $40.

There are also 1008 additional flops that contain two diamonds (and no aces). I'm still ahead, so I'm shoving, but he has 11 outs (8 diamonds and 3 aces) twice, or roughly a 44% chance of winning. I know this isn't exact, but working it out precisely (i.e., taking account of the hands that would result in straights for me but not him, or give me a full house, etc.) would be way too complicated.

Finally, there are 9997 flops that are dry for him and look safe for me--he misses his aces, his jacks, his straights, his flushes, and even his flush draws. In most of these, he will have just 3 outs (the remaining aces) twice, or approximately 12% chance of winning. Again, this is imprecise because there will be some straight draws unaccounted for here that might hit, as well as some runner-runner combinations. But, again, it isn't all one-sided, because some of these I'm lumping all together will also improve me even more than they improve him.

I'll spare you the arithmetic, and just jump to the conclusion: The average EV for this strategy works out to about +$180.


Strategy 3

Suppose I wanted to reduce variance even further, and not put the big money in when he flops a flush draw, even though I'm a favorite. That is, I fold in all of the situations listed in Strategy 2, plus every time two diamonds flop. Again, I'll skip over the boring math and just tell you the result I got: Average EV is +$165. It's obviously lower than the EV for Strategy 2, because I'm passing on situations in which I'm a favorite.

Interestingly, though, it's a slightly better strategy (though only by a small margin) than my pre-flop shove--if the assumption about Spewtard's conduct is correct.


Conclusion

If I could know for sure that Spewtard would put all his chips in on the flop no matter what came, then it would be smarter to wait for the flop, then either fold or shove according to whether I'm ahead or behind.

As I mentioned, this possibility did occur to me in the heat of the moment. I dismissed it because of the potential for losing the chance at stacking him if he were smart enough to fold upon missing the flop. That is a possibility, especially after I told him that I had seen his cards. But he just might have been so drunk and so uncaring about the money that he would shove regardless. The only way to know the true optimal approach would require being inside his head enough to know how he'd react. Unfortunately, we don't have that information. Because Strategy 2 is only about 13% better than Strategy 1 when we're making the assumption that he will shove on any flop, if he has much of any propensity to get scared and fold when he doesn't improve and I bet the flop, then Strategy 1 probably comes out ahead. It is, sadly, unknowable.

The way things came out, even if he had been willing to shove on the dry flop that came, I still would have lost to that damned ace on the river. My only consolation would have been that I got the big money in as an 88% favorite (i.e., on the flop when he had just 3 outs twice) instead of "merely" as a 68% favorite. I don't think that would have stung any less. Heck, it might have hurt even more.

But I'll have to admit, waiting for the flop before deciding whether to commit is a more interesting option than I initially gave it credit for being. Against a typical bad opponent, I'm confident that my approach was the highest possible EV. But against this particular bad opponent, I really don't know.

I find some consolation in this excellent paragraph from one of Matt Lessinger's Card Player magazine columns (which I have quoted before here):

I don't care if you are a rank novice or a world champion. It doesn't matter
whether you are in a tournament or a cash game. You could be playing for pennies
or Porsches. It's all the same. If you can get all of your money in as a 4-1
favorite, do it. And if you lose, live with it. It happens. Wait for
the next opportunity to arise, and then do it again. If you are able to
consistently create that scenario, you will be a successful player--end of
story.