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The Hand History
$2-5 at the Venetian
UTG+1 (covers) raises to 20, Hero calls in the BB with AQo
Hero checks, Villain bets 20, Hero calls
Hero checks, Villain bets 55, Hero calls
Hero checks, Villain bets 125, Hero calls, Villain tables KJ.
Can We Fold our Best Bluffcatcher?
First of all, is AQ no spade our best bluffcatcher? Our best bluffcatchers usually can beat some value hands and have excellent blocker qualities. We never have a better one pair hand since we are raising KK+ preflop. AQ is our only one pair hand that beats a potential three street value hand, KQ, and chops with another. Will we ever have two-pair+? The 7 and 2 don't improve anything from our likely flop calling range. The only stronger hands than AQ we may hold are KJ, particularly KJss. Those hands would raise river anyways, and although that counts towards our MDF, we are really just concerned with pot odds against a weaker opponent.
So what about blockers? We will probably have hands like QJ and some JJ OTR here. We can isolate those blocking effects by just examining a polarized range of hands that beat us with a few bluffs thrown in. Against an opponent who will never bet worse than AQ for value and has 3 missed spade bluff combos, QJ has 7.6%, JJ has 6.5% equity, and AQ no spade has 7.5% equity. Obviously, we should never call any of these hands against the given opponent, but interestingly having a Q is more important than maximally blocking the straight. Given the relative equality of QJ and AQ in terms of blockers, AQ no spade is clearly our best bluffcatcher when our opponent will sometimes bet AQ or KQ OTR.
To Exploit or Not To Exploit?
Whenever we fold our best bluffcatcher, we are making a highly exploitative play. In equilibrium, our opponent will want to bet in a way where we are indifferent to calling with most of our bluffcatchers and will likely slightly favor calling with our best bluffcatchers. That being said, it will often be correct to fold our best bluffcatchers against opponents who are severely underbluffing.
Against an unknown opponent, rather than dealing with a bunch of partial combos, it can be easier to think about a few versions of the same opponent. Sometimes a reg-type will never be bluffing here and never going for value as thin as KQ. Other versions of this opponent will go for thin value but rarely bluff. Others will rarely go for thin value but bluff often, and others will just be betting a lot. If we consider each of these four personas equally likely (probably not true, but easy to calculate), we can generate a composite to make a fairly comprehensive villain.
Against this villain who never bluffs or goes for thin value, we have no equity.
Against this opponent who goes for thin value and occasionally bluffs, we have 27.2% equity. Note that all bluff combos are spades only.
Against this opponent who does not go for thin value but will bluff at a higher frequency, we have 35% equity. Note that only about half of AJ combos are included.
Against the opponent who will bluff often and go for thin value (ahh, a villain after my own heart), we have 44% equity.
Against the composite villain, we have just under 27% equity. We are paying 125 to win 317, so we need 28.2% equity to call. There are lots of ways to fudge the analysis one way or another to say we have a slightly winning call or a slightly losing call. Against a reasonable range we should definitely be calling, but a high percentage of the field is drastically underbluffing here. I think when push comes to shove, this should be a tight fold against the field. That being said, you aren't making or losing very much money by calling of folding here, the definition of a marginal spot. Folding our worse bluffcatchers is a much easier exploit to justify mathematically.
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