Poker Musings, Vol. 2 - An Illustration of Poker as a Game of Skill

Question from one of the guys I play regularly with who was playing a 80 player multi-table tournament (MTT) at the track --

I had 11.8K chips when the blinds were at 1,000/2,000. With AQ I was sixth to act. The third guy to act limped in and everyone else folded to me. I decided to take a shot and I went all in. The limper, with about 14K chips, took a very long time but decided to call w/55. A king hit on the river but I missed my cards. I am still trying to figure out if he made the right call?

A typical poker situation here, but one that requires some thought. Not all of the variables are in accounted for (his image at the table, how the game was running, etc.) Here's a purely mathmatical response from moi...

"I think your opponent made the right call. He open limps 55, putting $5000 in the pot. That's usually a bad move unless the table was really passive, but that's not at issue here. You push $11.8K. It costs him $9.8K more to win $16.8K, for about what 1.7-1 odds? Is a call correct?

Depends on the range your opponent put you on. With an M<4, so less than 4 more rounds before being blinded out, your range of hands should be fairly broad. Hypothetically say you push with any pocket pair, AK-AJ, KQ, KJ and AT. That's not unreasonable I don't think.

There are exactly 6 ways to make any pocket pair (9c9d, 9c9h, 9c9s, 9d9h, 9d9s, 9h9s, for example). Your opponent's got an 80% chance to lose to 54 pocket pairs (AA-66), and 80% to win against 18 pocket pairs (44-22). He can tie with exactly 1 pair (if you have 55, there is only one way for him to have 55), but I'll throw that out because it clouds the water.

You have exactly 16 ways to make each of the overcard hands (AcKc, AcKd, etc.) So he's roughly 50/50 with 96 possible hands.

So call that (54+18+96)= 168 possible hands that it would be reasonable to put you on, pushing late-ish in an Multi-Table Tournament with an M<4. I might widen the range if you were first to act, but with a caller already I think the range is reasonably optimal. Additionally, calling really isn't an option for you, so if you are tempted to call you have to push. Your opponent probably knows this, in case you were saying to yourself "I'd fold AT." I don't think that's a good move against a single limper, so I'm including that in the range he could reasonably put you on. You might even be lighter than that-- e.g. QJ, QTsooted, JTsooted?. I'll start by sticking to the first range above.

So these are what appear to be your opponent's percentage of winning the pot. Study carefully and ask questions if you don't understand the percentages, 'cause this is important stuff to understand why he correctly called and you should too in his position (based on long term Expected Value and ignoring tournament chip vs. cash payout equity because I don't fully understand the math on that yet).

You have overpair. Opponent's chances--
80% loss = 54/168*(-$9,800)*.8= (-$2,520 expectation)
20% win = 54/168*(+$16,800)*.2= (+$1,080 expectation)

You have 2 overcards. Opponent's chances--
50% loss = 96/168*(-$9,800)*.5= (-$2,800 expectation)
50% win = 96/168*(+$16,800)*.5= (+$4,800 expectation)

You have underpair. Opponent's chances--
20% loss = 18/168*(-$9,800)*.2= (-$210 expectation)
80% win = 18/168*(+$16,800)*.8=(+$1,440 expectation)

Total expectation for your opponent calling with 55 against your range is $(-2,520 + 1080 - 2,800 + 4,800 - 210 + 1,440) = +$1,790.

So he can expect to make an additional $1,790 every time he makes that call making that firmly a + Expected Value move by him against that range.

But taking your hypothesis even further and "he should assume 50% of the time I'm on a higher pocket pair."

To get to that, let's say you call any pocket pair, AK, AQ or AJsooted. Again ignoring pocket 5's for you because they muddy the water.

As above, 54 pairs are ahead of your opponent, 18 are behind and 36 hands are 50/50 (only 4 sooted AJ hands). 108 hands total, 50% of which are an overpair like you said.

You have overpair against your opponent.
80% loss = 54/108*(-$9,800)*.8= (-$3,920 expectation)
20% win = 54/108*(+$16,800)*.2= (+$1,680 expectation)

You have 2 overcards against your opponent.
50% loss = 36/108*(-$9,800)*.5= (-$1,633 expectation)
50% win = 36/108*(+$16,800)*.5= ($2,800 expectation)

You have underpair against your opponent.
20% loss = 18/108*(-$9,800)*.2= (-$327 expectation)
80% win = 18/108*(+$16,800)*.8=(+$2,240 expectation)

Total expectation is $(-3,920+1,680-1,633+2,800-327+2,240) = +$840.

Your opponent is still +EV to call against the range you identified, which again is pretty darn narrow and most player's ranges will be greater than that in your position if they're pushing late with an M<4.

You seem to get caught up with there being 9 pocket pairs that he's behind, 3 he's ahead of and only 3 overcard hands, so 9>6 and he should have folded. It's just much more likely to be dealt an overcard hand than a pocket pair hand (16 ways vs. 6 ways), so it distorts what seems to be an intuitive bad call."