Another interesting quiz, no game theory needed

Just worked through a problem and was impressed with the answer. A buddy of mine who's a hugely successful limit hold em player and has been getting into pot limit and no limit asked the following:


$2-3-5 NL at Lucky Chances Saturday night (new lower-limit game!). UTG limps for $10. Sam B., a superb no limit pro, has the button and raises to $70. BB hesitates a bit, then goes all-in for $350 more. Assume BB is an average player. Fold to Sam, who has AKo. Should Sam call?


Now assume the BB will always go all-in with AA and KK. What range of additional hands must he also go all-in with in this situation to make Sam's call profitable?

He will have to call $350 more to win $850.


Let's lump AA and KK together. He'll be facing AA and KK 6 times out of the possible array of hands the BB has. Those 6 times, Sam will lose the pot about 90% of the time, giving him a return of $210 on an investment of $2100 . A loss of $1890.


So, the BB has to play an array of hands that will give Sam a profit of at LEAST $1890 on the rest of the calls. This comes out to be roughly: AK (a profit of $675 by calling 9 times and chopping -on average- an $850 pot), QQ (a profit of $450 by calling $350 6 times and winning an $850 pot about 3 times), and JJ (a profit of $450 same as QQ more or less). That totals $1575.


If you throw in TT assuming roughly the same profit ($450), Sam makes $2025 for all the times BB doesn't have AA or KK, for a meager profit of $135 on an investment of $11,550 ($350 for 33 hands) or a positive EV of $3.85 per hand.


Also, his "chop" with AK is actually slightly less than 50/50 because he only has AKo and sometimes his opponent will have AKs, giving his opponent a slight edge, but it's a little too thin to matter.


His total profit on calling with AKo is thus:


Opponent's hand range:

Down to TT: +$135

Down to JJ: -$315

Down to QQ: -$765

AA, KK, and AK only: -$1215

AA and KK only: -$1890

AA only: too ugly to think about.


In other words, this is a clear fold against an average opponent. Surprisingly, the call frequency should go up against BETTER opponents.


natedogg

Why should call frequency go up against better players?

I think its because there more likely to be trying something shady, instead of actually betting only when they have the goods.

Interesting how the breakeven point is just below JJ/AK. As Nate pointed out, if the player has any spunk to him, bluffing even 1% of the time, that'd be another 13.26 "hands" in which Sam is a strong favorite, drastically reducing the required span of good hands to more like QQ/AKs, which most players will play back hard with.


Sam called of course. Lost unimproved to a set of dueces.


Matt

mississippi gambler has part of it. It is because a good player will push all-in with a wide variety of hands considering the action and the stack sizes.


In no limit, when a real player raises from the button vs. a single weak limper, he rarely has anything. He wants the $20 out there and he's offering very bad odds to anyone who wants to play a marginal hand. He is also building a nice $150 pot to steal on the flop with his position.


So a real player in the blinds may often go all-in here with a wide range of hands (since he has only about $400 in front of him), sometimes with as little as nothing. The stack size here is just about PERFECT. Any smaller and it's not a good muscle play. Any bigger and it's too much exposure just for the $70.


It's also because you, the button raiser, know that the big blind is a real player. A real player knows his opponents well, and therefore he knows you well, and he knows you might be raising with as little as any two cards. You know he knows this about you.


He may be going all-in with anything, just because he knows you would raise here with almost anything, giving himself a VERY nice price on the $70 you just threw in there. He knows it's highly unlikely you have a hand even as good as AK, much less TT. You will have to lay down the vast majority of the hands you could be raising with here.


Therefore, his all-in bet of approximately 5x the pot is most likely a money-winner.


But now you get into the situation where his hand range could be so wide you might have a profitable call with AKo. In fact, you almost certainly DO.


Take for instance the time I raised 300 preflop and called 1100 more with AQ and it was good (until the flop of course). Damn you Matt Flynn, with your ten high button bluff. And the luck of the Irish to go with it.


A word of advice to anyone who made it this far, if you're going to bluff all in with ten high, make sure you get lucky on the flop.


natedogg

Interesting! I would not have thought an average player would go all in against a very good player with only 22. You would think an average player, as natedogg pointed out, would be unlikely to make a play like that, especially against someone he knows to be a good player. Maybe he's not so average!