CardSharp

OK, it’s time for the answers to the first hand quiz.  I had hoped to produce a little more discussion, but I think making it 7 card stud scared everyone off.  That’s unfortunate, but I’m not easily deterred.  So I’m going to keep doing hand quizzes.  I do think in the future I’ll make most of them holdem hands though.

Now, the answers (out of order):

Question 1/Question 4:

First off, let’s get one simple issue out of the way.  If you bluff, you MUST bluff both 6th and 7th.  Villain’s call on 4th indicates that’s he’s willing to pull from behind in this hand.  There’s no reason to think he’ll change his mind on that point on 6th.  So betting 6th but not 7th is just throwing money away.  Similarly, betting 7th but not sixth won’t work – your bet on 7th won’t be plausible if you checked 6th.

Now to the game theory.  The nature of game theoretic optimum strategies is that they make your opponent’s decisions not matter.  In this case our opponent is going to have two possible choices – call or fold.  Raising would be a horrible play for him.  In order to find the point where villain’s decision doesn’t matter, we need to do a bit of pot size and odds math.

On 6th street, the pot is going to be roughly 5 1/2 big bets depending on the exact size of the ante.  Although not explicitly stated, you of course double-bet 4th.

If villain decides to call, he risks 2 big bets to win 7 1/2 just by simple pot math?  But what about the draws?  Both you and villain potentially stand to improve on the river with fairly live draws.  This is where the simplification from Question 4 comes in – the easiest way to proceed is to simply to assume that the draws both directions cancel each other out.  That’s not perfectly true, but it’s close enough.

Now back to the game theory.  In order for villain’s decision not to matter, you must be bluffing two times for every 7 1/2 times you have the boat.  Which brings up a good question: given the betting line you took, how frequently will you actually have the boat?

It’s pretty clear from the betting line that you had SOME pair to open.  But was it split or buried?  A priori, 1/3 of all pairs are buried and 2/3 are split.  If we use that number, you would want to bluff roughly 1/2 the time when you have the buried pair (and thus 3 pair) since that would leave 1 bluff for every 4 non-bluffs, which is just about the odds that villain is getting and thus renders his decision meaningless.

Question 3/Question 2:

There’s a bit of a problem in my logic above.  While, to begin with, it’s 2:1 against hero having a buried pair, there is evidence in the subsequent play that indicates he has one.  If he really had split 2’s, would he be eager to call a pair of kings without himself holding an A kicker?  Probably not.  And aces are 1/2 dead.  Even if hero did have {A 2} 2 he probably wouldn’t call because his kicker was so dead.  This is in essence a Bayesian approach to the problem – hero’s willingness to play “changes” the odds that he actually has deuces vs. the odds he had a buried pair, in this case radically in favor of the buried pair.

Now, here’s where it gets tricky: if you think villain is going to adopt such a Bayesian approach, and conclude you didn’t actually call with split 2’s, then you must adjust your bluff percentage WAY downwards – essentially to zero.  But it all depends on your opinion of villain, which gets to the answer to Question 2.  There are basically three types of villains that are relevant here.

1. “clueless guy” – this guy is going to ALWAYS call 7th when he can beat your board.  He doesn’t understand Bayes, but he doesn’t fold at a game-theoretic optimal rate either.  You’re better off with an exploitative approach against him, which in this case means never bluffing.
2. “tight old dude” – one oddity of 7 card stud is that it hasn’t been a popular game, except in certain small enclaves, for 20 years.  That means a disproportionate number of the players are elderly, and a lot of them play INCREDIBLY tight by modern standards.  In the real hand I drew this quiz from, villain was one of these guys, and I actually did bluff.  And he actually folded.  I was amazed.  If your villain has already made some really tight folds based on reads early in the session (especially if those reads were wrong), I’d typically bluff.
3. “stud genius” – anyone deeply familiar with stud will pick up on the Bayesian inference that you don’t actually have deuces full, and will call 100% of the time.  Don’t bother bluffing that guy.

So in reality, there’s only one type of villain I’m likely to ever bother bluffing, and I’ll bluff him way more than game theory says would be correct.  Everyone else I’ll check to.  As I’ve said before, I typically avoid game theoretic strategies, although it’s important to know what they look like.  This case is no exception.