What’s Your Bluff Target? Results

Thanks for all the comments on What’s Your Bluff Target? A few people disagreed with my premise that a bluff of some sort is mandatory here. I think that Samael sums up the argument for betting something quite well: “we have absolutely no sd equity, and the villain has lots of air and two paired hands that are vulnerable to three streets of aggression”. For those who still aren’t convinced, I recommend my Bluffing series on Tournament Poker Edge.

The fact that Hero has a lot of air in his range for getting to the river because of all of the missed draws is not an argument for never bluffing. It may be an argument for bluffing rarely with small sizing and extremely weak hands, such as this one, simply to deny a showdown to the weakest hands in Villain’s range. It may also be an argument for betting really big, such that it’s hard for Villain to call even if he knows that Hero has a lot of bluff candidates.

I think there are three options to consider given Villain’s likely range:

1. Bet small, anywhere from $50 – $250, targeting missed draws that have not paired the board.

2. Bet a medium amount, $200 – $500, targeting two-pair hands like A9s, 98s, TT, and JJ.

3. Overbet the pot, $500 – $2000, targeting trips.

Exploitively, I like mwalsh’s process for thinking through these options: “we proceed bluffing a higher and higher amount until we reach a category of hand where we do not believe attempting to bluff that category of hands adds marginal value”. If, for instance, we think that Villain will fold to a shove 100% of the time, then we can exploit that by shoving, and even if a small bet is +EV it will not be as +EV as shoving our entire range. Note that in such an extreme case shoving even a Q with a bad kicker could be correct.

Betting Small

This is a case where size matters, because Villain has missed draws that could consider calling a river bet getting a very good price. Many hands in this category will contain an Ace or a King, and I think that betting $50 will not be enough to shake them.

It’s worth thinking about what you’re representing when you make this bet. Hero would presumably raise TT and JJ pre-flop. This leaves the bottom of Hero’s value betting range at 9x (ambitious, but not out of the question considering how much air is in Hero’s range) and Qx (note that Hero’s kicker can’t be too strong here as AQ and probably KQ are raising preflop). The only 9x combos in Hero’s range would also be flush draws, so there aren’t too many of those. Thus, the value hand to balance around for this bet size is really trips with a kicker that can’t expect to be ahead of Villain’s Qx range. More on this in a moment.

Betting Medium

Brian correctly points out that “77-JJ are all pretty heavily discounted from his flop overcall if this is truly a nit”. The other plausible two-pair candidate with be 9x of diamonds, of which Villain can’t have more than a few candidates in his UTG limping range. If you’re going to target just one hand type, it probably isn’t best to pick the one that represents the smallest part of Villain’s range. I’d also wonder, for exploitive reasons, whether Villain is going to fold these hands to any “normal” bet size on one of the best river cards in the deck after coming this far with them.

It’s interesting to note that there’s some overlap between the bet sizes I suggested for the small vs medium bluffs, and that’s because AxTd is not so different from Td9d. There may, then, be a case for trying to find a bet size that targets both of these regions, which would probably be in the $200 – $250 range. This is consistent with how I’d play a Q, and I think it gets enough folds from unpaired hands to show a profit. If Villain chooses to fold any pairs, that’s icing on the cake.

Betting Big

Many commenters seemed not even to consider the possibility of Villain folding trips. With about 4x the pot left in the effective stacks, that’s a mistake. Calling a shove or even a 2xpot bet with AQ is not a trivial decision. As Brian puts it, “Threatening $1000 of his hard, nut peddled dollars should get him thinking pretty quickly.” I suspect that many of you who don’t want to try to bluff him off of trips wouldn’t actually shove $2000 here with sixes full, and you should ask yourself why not.

One of the few cases in no-limit hold ’em that is completely solvable by game theory is when you hold a polarized range in position on the river against a range that consists purely of bluff catchers. The correct play is to shove all of your nuts hands and enough bluffs to make your opponent indifferent to calling.

Say that the pot is $500, Villain has exactly $2000 behind, and we know for a fact that he has KQ. Facing a 4xpot shove, Villain would have to win 44.4% of the time to break even on his calls. Thus, Hero should be bluffing 44.4% of the time that he shoves. If Hero’s value range consists only of flopped sets, that’s six combos, so we need 4.76 bluffs to balance that.

Let’s compare the EV of this strategy to that of a more common strategy, which is to bet a “safe” amount – say $400 – that we’re sure he’ll call with all of our full houses and never bluff. If he really does pay this bet off every single time, then you make $900 with your 6 value hands and lose every time with your 4.76 bluff candidates, so your overall EV with these 10.76 hands is $2400 $5400/10.76 = $501.86.

If we were to shove 4xpot with those same 10.76 combos, then we are indifferent to whether Villain calls always, sometimes, or never. Thus, by assuming that he always folds, we can see that the EV of this strategy is $500, the size of the pot. There is nothing Villain can do to prevent us from stealing the pot with those 4.76 bluff combos, because he’s going to run into our full houses just often enough when he calls.

Although the EV of these strategies is practically identical, the significant thing about the shove is that it’s not at all sensitive to our assumptions about Villain’s play. If it turns out that we’re wrong and Villain does occasionally fold KQ to the $400 bet, then we would have done better by shoving a balanced range.

Of course my example is a little contrived. If you believe you can reliably get a bigger bet than $400 paid off, then the exploitive strategy of only betting big for value will perform better than the GTO shove. I didn’t have that confidence in-game, and I suspect that anyone putting so much confidence in inferences drawn from a paragraph or two explaining some broad dynamics is making a mistake.

The whole thing gets a lot more complicated if we put slowplayed boats into Villain’s range, because at that point we’re no longer shoving into a range of pure bluff-catchers. I do, however, think there’s sufficient reason to rule those out more or less completely. Sets and two-pair are extremely unlikely to overcall on a draw-heavy flop, we’ve seen this Villain fast-play in a questionable spot before, and he probably isn’t limping the Qx combos that would make Queens full anyway.

You can find more discussion of this and more complex situations in The Mathematics of Poker.

Two Bet Sizes

Because an optimal river shove employs only 4.76 bluff combos, Hero probably has enough bluff candidates to balance an additional, smaller bet size. Thus, it may be feasible both to shove the river with full houses and 4.76 bluffs and also to bet something in the $200 – $500 range with the rest of our bluff candidates plus our Qx (to find a GTO size, we’d have to look at the ratio of value to bluff candidates in our range).

The question, then, would be whether 7h 5h belongs in the medium bet or the big bet range. It’s unique among Hero’s bluff candidates not so much because it has about as little showdown value as possible but because it doesn’t contain any diamonds, which are actually pretty significant blockers to his folding range when we make a medium-sized bet. For this reason, I think that this exact hand is better used to balance the medium-sized value bet, and the 4.76 bluffs that get shoved should be chosen from among Hero’s busted diamond draws.

Results

I bet $800, and Villain quickly folded. As argued above, I don’t think there’s a lot of reason for this intermediate bet size. I think in game I was allowing for the possibility that I might occasionally run into a slowplayed full house, but looking at the hand now I really don’t think that’s going to happen. In any event, I think that 7h 5h ought not to be one of the candidates for bluff shoving, so $200 would have been better.