Wednesday, August 31, 2011

Poker Quiz Question #26

Q#26: You're in a SnG. Blinds are T100/T200, and you have T1000 in chips. Everyone else at the table has approximately T5000. You're in middle position and it is folded to you. You hold Th-3s. What should you do?
  1. Fold
  2. Raise
  3. All-in
  4. Call

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A#26: Assumptions: First, because our M is 3.3 (or 5x the big blind) and we're the shorty at the table, let's assume we're in push or fold mode. In other words, I'm going to summarily eliminate Calling and Raising (less than all-in) from the list of possible answers. Therefore it boils down to whether we can profitably push or not. One way to answer an all or nothing question like this is via a simple EV calculation.

Reads: Let's assume this is a nine handed table. We're in MP, so that means we have ~5 players left to act. We don't know anything about the remaining five players except they each have bigger stacks than us, they each hold ATC, they're fighting to stay at least even with each other, and they know we're getting desperate.

Estimate: Th-3s is a fairly weak hand; if we run 'stove against 5 random hands left to act, our equity is only 11%. This sounds bad, but it's not quite the end of the story. We do have some fold equity, as our T1000 represents 20% of any other player's chip stack. If we run our hand against a single player in 'stove with a villain's likely calling range of, say, all pairs bigger than 55, plus the bigger Broadways, we're a 75:25 dog.

But this calling range represents just 8.9% of hands dealt to an opponent. The chance that one of the five players downstream of us has one of these 8.9% hands is: 1-power((100%-8.9%),5) = 37%.

So, 37% of the time we get called and, when we do, we win 25% of the time (T1300 added to stack) and bust out the other 75% of the time (T1000 lost from our stack). The other 63% of the time we just take down the blinds uncontested (T300). Our EV is therefore:

EV=[0.37 x ((0.25xT1300)-(0.75xT1000))] + [0.63 x T300]

or, EV = +T30

This was an unexpected result for me when I ran the numbers; I fully expected the answer to be negative, but it's actually slightly positive. For fun, I decided to play around a bit with different opponent calling ranges and see how the resulting EV changed. Here's what I got:

The horizontal X-axis is the range of hands that our opponents call-off with, and the vertical Y-axis is the resulting EV for us. As you can see from the graph, the wider the range that your opponents are going to call with (and/or the more of them that you have to get past), the less inclined you should be to shove with weakish hands like T3o in this situation*.

Decide: Given that we are +EV at T30 (and we're desperate), I think we're on a (semi)Bluff line here.

Implement: Shove and pray.

Answer: All-in

All-in for now....
-Bug
*The graph goes through zero at around 11%, which per 'stove represents a villain calling range of: 77+,A9s+,KTs+,QTs+,ATo+,KQo. Qualitatively, this range passes my giggle test for hands I would personally call a shove with if I were one of the villains in this hand.

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