Cracking

Here's the situation.

You're in middle position with $150 in a 9 handed online 100NL game, you get all five limpers to your pocket eights. You decide to limp as well.

The player on your immediate left in the cutoff has $200, is pretty tight, makes it nine to go. It folds around to you. You believe the raiser is not a moron or maniac, probably an above average player for this level, quite tight, preflop raising about 4%, reasonably aggressive post flop.

What do you do?

Easy decision, fold.

Or is it?

Two relevant questions:

1. Can you put this player on a hand?

Question one, most likely yes. They really don't want to see a flop with multiple callers and the number of limpers requires a big bet. This player is very unlikely to raise here with worse than queens and perhaps not with queens--I've found many tight players facing this many limpers with something like TT-QQ will often either play it for set value or overbet to suss out an overpair or just take it down preflop. A slightly more aggressive player you might include TT-JJ and AK, but let's say you are confident it's about 40% aces, 40% kings, 20% queens.

2. Will they pay you off if you hit?

Several factors here.

You have about a 1 in 7.5 chance of flopping your set.

If you hit when an ace, king or queen falls as well, it's going to get really tricky and you're going to have play much more carefully. Even in these cases you're still way ahead 60% of the time if an A or K has fallen and 80% if a queen has fallen, but your EV after the flop is significantly reduced as you've gone from basically 92% chance of winning to something more like a weighted 70%. This will happen approximately 40% of the time that you hit (assuming he has AA-QQ, there are 10 cards that will be threatening on the flop, with two shots assuming the third is your set). To simplify things, let's just assume that you can only hit "clean" sets but that they only hit 1 in 8.5 times (please feel free to chime in if you think this simplifying assumption is unwarranted).

Let's assume your strategy is to make a pot sized check-raise if you hit, check-fold if you miss.

If you hit and your opponent bets pot, there will be 37 dollars in the pot. Here's the crux of the question. If your opponent is capable of folding here, you made a mistake by calling preflop. Here's why: you paid 8 to win 29 and you only had 1:8.5 odds to do so. To make the call correct, your going to need him to call at least 39 more--even if he calls a minimum raise of 18.5 and then folds to a turn bet, you're still not getting the right odds, especially when you consider he does have two outs.

So let's say you pot it to 55.5. If he calls on the flop and folds on the turn to a push if he misses, you'll be getting 84.5 - 6 (4% of 150 of you losing your stack if he hits on the turn), so 78.5, which is makes your payoff 9:1 and thus good. If he pushes/calls and calls a turn push, you'll make 148 - 12 or 136, which is a pretty sweet 17:1 reward. (I think actually players that make that first call are probably going to make the second one as they are pretty well pot committed, so more likely than not you're effectively going to get the folder or the raiser--in fact the "better" players making this call will almost always push.)

Further breaking it down: the EV of against a 100% folder is approximately -5 (27/8.5 - 8), against a 100% caller is +1.5 (78.5/8.5 - 8) and against a 100% raiser is +8 (136/8.5 - 8).

In most cases you'll be unable to predict the response 100%, but I think this move can be profitable against many players you face at 100NL and below, especially if your "solid" opponent doesn't think you're so solid (if you don't have an aggressive image, this is going to be less effective). I don't have enough experience to know if this holds at 200 NL or above. Also, if the stacks are smaller or bigger, the math will change of course, though I suspect there is some balancing factors in either direction (for example, bigger stacks means bigger rewards but probably a lower chance of being paid off with a whole stack and vice versa).

Time to get cracking?