One of the more interesting aspects of mixed games is having to deal with a variety of characters at the table, each with their own style, strategic outlook, and motivations.
While it’s important to have some knowledge on at least an approximate set of game theory optimal solutions, we are mostly looking to exploit obvious tendencies of our opponents while playing in a manner that won’t give anyone else the idea that they can possibly do the same to us.
Example
Let’s walk through a somewhat common situation in Deuce-to-Seven Triple Draw Lowball where we might vary our play based upon our opponent, where part of the reason for doing so is to avoid potentially being taken advantage of in future hands.
Suppose we open from the hijack position with a premium one-card draw (2-3-4-7), our opponent called on the button with 2-5-8, and the blinds fold. On the first draw we failed to improve our hand, and our opponent called our bet on the flop having picked up a one card draw of his own, 2-5-6-8. On the second draw we catch a ten to make 10-7-4-3-2 and decide to check the turn.
Betting with the intention of patting when just called would have been sub-optimal since we don’t know the outcome of our opponent’s second draw, and getting “frozen” would be a bad outcome with such a premium draw underneath.
For instance, if our opponent makes an 8-7-6-5-2, he’s probably just going to call hoping we pat a worse hand, ensuring himself the pot. We could also get raised by 8-6-5-3-2, in which case we are putting an extra bet as a sizeable underdog.
Making A Decision
However, in this circumstance he failed to complete his hand and checked back, intending to draw. Of course, we never know precisely what our opponent has, but for the purpose of this discussion, assuming perfect clarity allows us to fully focus on the principles in play.
With one draw to go 10-7-4-3-2 has 70% equity against 2-5-6-8. 2-3-4-7 has around 54% equity for a difference of approximately 16%.
Given this, patting would be a reasonable play, and with 3.75 big bets in the middle, keeping the 10-low “preserves” around .60 (16% * 3.75) of a big bet. However, the hand doesn’t end on the turn, and we should check the river hoping we don’t end up facing a bet.
Evaluating The Situation
On the river, do we want to be up against someone who never bluffs against a somewhat face-up 10 or jack low? Or is it preferrable to be playing someone who bluffs around 50% more often than they should?
Against someone who never bluffs, our expectation is 2.63 big bets (3.75 big bets * 70%) as we have a 70% chance to win and can safely fold to a bet.
We can now attempt to estimate our expectation versus the “over-bluffer” who we will assume is betting the following 18 cards:
Value (13): Threes (3), Fours (3), Sevens (3), and Nines (4)
Bluffs (5): Deuces (2), and Eights (3)
What cards our opponent chooses to bluff with really isn’t that meaningful as we obviously have a made holding that can beat any pair. However, many players opt to bluff a pat hand when pairing deuces, and here we’ll assume he also can’t help himself when pairing an eight.
Since the pot is laying 4.75:1 (target win percentage 17%) we make the call as we’ll take the pot around 28% (5 bluffs/18 total betting cards) of the time.
The Hidden Cost
While this is the right decision, there is a cost associated with picking off the bluff as you lose a fraction of a big bet when making the call. That’s because 72% of the time we are losing that final bet for a net loss of around .44 of a big bet.
When our opponent checks, we are going to win the 3.75 big bets in the middle. However, when he wagers, this loss on the river betting round lowers our overall expectation. The expectation against the “over-bluffer” when reflecting the reduced frequency that the river gets checked through is around 2.43 big bets, which is .20 less than the 2.63 versus someone we can exploit by folding.
Perhaps this is intuitive to most, as the “non-bluffer” allows us to essentially be “all-in” on the turn and we fare better taking advantage of that tendency as opposed to playing against someone who errors by bluffing a bit too much. In theory, someone is playing perfectly if they bluff at the exact frequency to make us indifferent between calling and folding.
From the viewpoint of the in-position player it’s desirable for his image to be aggressive enough that he can under-bluff this particular situation and still collect bets. For example, if he somehow constructs his value to bluffing range in a manner where his opponent has only a 10% chance to win, he’s doing quite well when getting a call on the river.
Making Sense Of The Numbers
Admittedly, all of these theoretical considerations can get confusing and make one’s head spin. In practice, it’s highly probable that the overall population generally under-bluffs this scenario, and against the majority of opponents your hand might not be best enough of the time to make the call.
But it’s somewhat annoying in that you could sometimes be folding the best hand, and observant opponents might think that they can possibly take advantage of you in the future. After all, your hand is relatively face-up, and if you fold in this spot to a somewhat aggressive opponent, an onlooking tight player (who you may have pegged as someone who wouldn’t bluff) may surmise that he ought to also get away with it should the opportunity arise.
Therefore, unless you are up against someone you are confident never bluffs, it might have been best to have broken the 10-7-4-3-2 to begin with. While you are taking an equity hit and are out-of-position, your premium draw can expect to recoup some of the shortfall on the river, and on your good days might get several bets in when making the nuts.
It’s also a lot more fun than check-folding a 10-low, and sometimes you can further tilt yourself if you decide to rabbit-hunt and discover that you would have caught gin.
Kevin Haney is a former actuary but left the corporate job to focus on his passions for poker and fitness. The certified personal trainer owned a gym in New Jersey, but has since moved to Las Vegas. He started playing the game back in 2003, and particularly enjoys taking new players interested in mixed games under his wing and quickly making them proficient in all variants. Learn more or just say hello with an email to haneyk612@gmail.com.
