Risking the Pot for Profit

In a small online pot-limit hold'em game recently, holding A 7 in early position, I flat-called, expecting to see the flop for my dollar. Five other players had the same idea, a pattern common in the small no-limit and pot-limit games online. Many of the players in these games, like myself, have little experience with big-bet poker outside of tournaments, and like to wait for the flop before making a commitment.

The flop came A 6 2. Top pair, bad kicker, and a backdoor-flush draw. I couldn't remember why I was in the hand, so I checked. A middle-position player opened for $2, a late-position player called, and I called, as well. I didn't feel particularly attached to my ace, but I was happy to take off a cheap card. I had the sense that the other players would be likely to pay me off if I somehow made a hand. I'm sure they had the same sense about me.

The turn was the 9, bringing me a flush draw. I bet the pot, all of $12. The original bettor folded and my lone remaining opponent raised $12. At that point, the pot offered me 4-to-1 immediately on a call, plus (or minus) the possibility of further action on the river. With odds of 4.11-to-1 against making my hand (and 4.75-to-1 against making the nuts), I decided that was good enough, since at worst I could fold anything but a flush, and I thought my opponent would at least consider calling if a heart hit the board. So, I called, made my flush, and bet $36 into him (his remaining stack) on the river. He called, and I went home happy.

My opponent, not surprisingly, was unhappy, and showed the set of sixes he'd flopped. He chastised me for calling his raise on the turn with a flush draw. "How could you call on a draw?" he typed. Was he sincerely upset? It's hard to tell online. There's really no reason to underbet the pot there if you'd really like your opponent to fold. But maybe he just clicked the raise button without thinking, or had put in advance action, thinking I was going to check. I was inclined to view his raise as an error, and although I don't tend to chat online, I was thinking something along the lines of, "How could you underbet the pot that drastically?" Clearly, I thought, he should have bet at least enough to make it incorrect for a big draw to call. By raising only a third of the pot, my opponent either intentionally or otherwise gave me terrific odds to hit my flush, and effectively made my call correct.

Or did he? I didn't know he had a set, after all. I might have been inclined to pay off a small bet on the river if a blank hit. I might have been inclined to pay off if I paired my kicker or made trip aces. I don't know how good a read my opponent had. But it's worth figuring out his expectation from his point of view, to get a better sense of whether or not his action on the turn made sense.

To review, here's the situation:

Me: A 7

Him: 6 6

Flop: Au 6 2

Turn: 9

As usual, I'll have to make a few simplifying assumptions, just to keep the problem tractable. Let's assume my opponent knew I would fold to a pot-sized raise, but that I would call his underbet (and that $12 was the largest raise I would call). Let's further assume that he knew all of his chips were going in on the river no matter what (a fair assumption, since the river was his worst card and he still called). The remaining question in his head would be how likely I would be to call, depending on what came on the river. Assuming he had a pretty good idea of what I had, we can divide the 46 unseen cards into six categories:

We can evaluate our respective expectations, or EVs, from the point where I've bet on the turn and he's contemplating how much to raise. If he raises the pot, it's easy, since we're assuming I fold to that bet. His EV is the pot prior to his raise, or $24, while my EV is zero.

If he raises a third of the pot, $12, and I call, his EV depends on how I will react to different river cards. For each winning card (for him), his EV is $24 (the existing pot, including my bet) plus $12 (my call on the turn) plus $36 times the probability that I will call or bet the river. For each losing card, his expected loss is $24 (his entire raise on the turn) plus $36 times the probability that I will call on the river (which I always will). If we weight these different contributions by the probabilities above, we'll have his expectation.

The $26.97 is more than $24, so, clearly, the underbet was preferable, at least from the point of view of someone who made those same assumptions about how often I would call. It's a close call, and even in a game of this size, a profit of $2.97 isn't a huge coup. But the bottom line is that the likelihood that I will call his river bet with a losing hand outweighs the risk associated with his giving me a cheap shot at a flush.

Although it's a trivial exercise, we also can reassure ourselves that my calling his raise on the turn, which I regarded as a no-brainer, was a mistake (from the abstract point of view of someone who can see all the cards). The size of the mistake should be the same size as what he gains when I call his small raise. Again, from the starting point before my call, my EV when I win is $36 (the size of the pot before my call) plus $36 (the amount I'll get from him on the river), or $72. My expected loss the rest of the time is $12 (my call on the turn) plus $36 times the probability I call on the river.

His $2.97 profit is my $2.97 loss. It's nice when things add up. As above, I'm done in mostly by my disposition to check and call the river with some losing hands. Although the play looked profitable to me, that's because I assumed there was at least a chance I could win without a flush. It's worth noting that even if we assume I'll never call the river with only one pair or if the board pairs a card other than an ace, it's still break-even for him (in fact, it becomes exactly break-even). From his point of view, my calling the small raise is preferable to my folding, unless I can also dump some of the other weaker hands, such as when I pair my kicker or make trip aces. If he thought I was likely to do so, his underbet was probably an error.

Of course, it doesn't follow that my call on the turn was incorrect, or that I really should fold two pair on the river, any more than it's incorrect to call an all-in bet from a maniac while holding K-K when he happens to hold A-A. When you have an unusually strong hand, you often want your opponents to play subjectively correctly. From my point of view, he might have had a range of hands, and his set was the worst case. If I'd known he had a set, of course my call still would have been correct, but only because I then would have been able to change my behavior on the river, and bet my flush only when the board didn't pair. Since I couldn't see his cards, I couldn't completely avoid paying him off on the river. He was in a much better position, since my actual holding was probably quite close to what he would guess. And if he was wrong about my cards, there were few opportunities for him to make big avoidable mistakes.

This analysis is obviously oversimplified in many respects. For example, he also might have assigned some probability to my having various other hands, such as a single pair with no flush draw. Perhaps he should have been worried that his underbet on the turn would tip his hand, and allow me to play closer to perfectly. We also haven't considered the impact of other possible betting patterns on the river. And there are several other decisions in the course of this hand that could stand some scrutiny. Even with a raft of assumptions, it's impossible to do all this arithmetic on the spot. But running through these exercises afterward is good fodder for our intuition. As I've said before, it's usually more valuable to play with the numbers and try out different assumptions than it is to plug in a single set of numbers and go with what they tell you. In working out the results for this column, I used a simple spreadsheet. I think the results I've presented are fairly typical of what you'll find with reasonable assumptions. But, you might disagree. I'll make it available on my website (http://www.seriouspoker.com/), in case you want to try it out.

After the hand, my opponent was probably kicking himself for an underbet that cost him the whole pot. I was feeling a little smug after taking advantage of his lapse. But to an omniscient observer, his underbet was the best way to exploit the fact that I would, quite reasonably, underestimate his strength on the river. Though probably accidental, it was most likely the best play. My call, though subjectively correct, was a negative EV move given my opponent's cards. He should have been pleased to get the call, even if it didn't work out, and I should have been relieved to have dodged a bullet.