One card

No new poker here as we had a get together last night at Filadelfio's a local, slightly upscale, neighbourhood bar and grill, which actually does a pretty good job of emulating the real thing (just to choose an example I happen to be familiar with). One item on the pizza menu there, is a ‘Canadian’ pizza. Now what toppings do you suppose that has? I almost don't want to spoil the suspense, and force you to follow the link instead, but I'll give in: salmon, sour cream, gherkins (that is, dill pickles, for the linguistically challenged). As many poker players have so eloquently put it in the chat boxes of a myriad of poker sites: WTF????? I have a peculiar fascination with this pizza, rather like Steinbeck's character Doc in Cannery Row who was obsessed with the idea of a beer milkshake. Some day, I suspect, I'll just have to break down and have one.

Rooder left a comment on yesterday's post (thanks for that!) which leads me to suspect that I didn't make the point I was trying to make as effectively as I might have (alas, it was ever so). I was trying to fight my training and not present a simplified example (as in the mathematician of joke number 6) but, in light of the comment, and a failure of any other sort of inspiration, here we go.

Dad puts $1 on the table, and also gives $1 each to his twin sons Billy and Bobby. He then shuffles the 13 hearts from a standard deck of cards, and deals one card to each boy. The game is ‘One card hold'em’. Billy can bet $1, whereupon Bobby can fold or call; or Billy can check, whereupon Bobby can bet or check, and if he bets, Billy can choose to fold or call. In the event of a show down, higher card wins (ace is high).

Billy, who must act first, receives the J♥. He must decide to bet or check. Since Bobby is his twin brother, Billy knows how Bobby will respond to each action. If Billy bets, Bobby will fold any card 8 or under, and call with 9 or better. If Billy checks, Bobby will bet any card 8 or better, and also with a 2, as a bluff. No claim is being made that these strategies are in any way optimal, just that they are what they are.

Under these conditions it is clearly correct (do the math yourselves!) for Billy to play passively. The short reason is that Bobby is betting with more hands than he's calling with, and the extra hands that he bets with are ones that Billy beats.

The real hold'em situation was meant to mirror this. The key ingredients are: heads up play; a good hand, which is a decent favourite against your opponent's range; a very dry board that gives your opponent few outs if he's behind; and an aggressive opponent who is likely to bet more hands than he would call.

One purpose of aggressive play out of position at hold'em is to buy information. If the information given by a call is something you would rather not have paid for, it might be time to consider playing passively instead.

Edit: Oh yes, in the real hand, I was the TAG player in MP and held AK. I might have chosen to check the turn, since I guessed that I was well behind to a pair of jacks, but there are a lot of players out there who routinely call apparent continuation bets with air, but fold to a second bullet. Further, I thought that there was some chance I might actually fold a jack worried about an overpair, or more reasonably a middle or bottom pair hand (people do call raises from the BB with Ax sometimes). BB's call on the river shows that he wasn't thinking along the lines I've been describing, but it occurred to me as I went over the hand (as I tend to do when I've put nearly half my stack in behind) that up to that point, there was little to criticise in his play.

Schrödinger's hole cards

I described the following hand in my previous post. Despite the underwhelming response, I'd like to share some thoughts about it. The short version is:

Against an aggressive opponent, it may well be correct to play top pair, top kicker type hands from out of position quite passively.

The underlying reasons for this remind me of the superposition of states that arises in quantum mechanics, most generally illustrated by means of the example of Schrödinger's cat (which, oddly enough, was introduced as a thought experiment to show the ridiculousness of the macroscopic interpretation of superposition of states.)

In the example hand, the BB could, at several points make an ‘observation’, which would collapse or partially collapse the ‘wave function’, consisting of MP's unknown cards. For good reasons, or otherwise, he does not do so -- for to do so would leave him in a less advantageous position than playing against the mixed state.

Cash game, NLHE, both start with about 100BB.The big blind, holds A♦J♦. A tight aggressive player in MP opens with a standard 3.5BB pot sized raise. BB calls, and the two players take the flop heads up.

Hard to argue with this decision. Even consigning the small blind to the rake, BB is getting 9:5 on the call. Flops with an ace could be tricky, but flops with diamonds or a jack should be good.

Flop is J♥4♠2♣. BB has not seen MP play a huge number of hands, but MP has always taken action on the flop playing heads up after being the preflop aggressor. BB checks.

The alternative is, of course, is for BB to bet out. BB is currently a significant favourite against MP's range (about 60:40 against a conservative TT+, AJ+). The difficulty with betting out is that it allows MP to ditch most of his dominated hands, but to call or raise with JJ+ and enough semibluffs to make future life difficult. BB does not want to reduce the hand range that MP might be playing here, as MP folds will always be right, and MP calls/raises will or should be judged well enough to put BB in a pickle. I don't think that check can be said to be ‘wrong’ here, it's a defensible option.

MP bets 5BB, about 2/3 of the pot, which has been his standard bet in this position, sometimes as a continuation bet, sometimes with strength. BB calls.

According to the previous comments, MP's action does not limit his range in any way. Against that range BB is a favourite, so a call (at least) is called for. What about a raise? Some of the previous comments still apply. At a guess, MP will take the raise to show AJ (maybe some other good jacks), JJ+, 44 or 22. Suppose that a raise will knock out everything except JJ+ and AK. BB is actually an underdog against that range -- so that anytime the raise is called, it will be called correctly (allowing for the fact that from BB's position, the call only indicates a range of hands), and more or less every time the hand is folded, it will be folded correctly. It can be, indeed frequently is, the case that both one player's raise or bet, and another's call are correct, but here the continuing difficulty for BB of playing the hand out of position tends to argue against it.

The key here is that BB could have adopted the following strategy immediately after the flop: “So long as MP's actions don't significantly affect his range, and so long as low cards continue to hit, I will always call.” Based on his assessment of what MP's range is, that strategy had positive expected value, and the more MP bets, the greater the value becomes.

Turn is 6♠. BB checks. MP bets 10BB at the 17BB pot. BB calls.

The above comments still apply. It would be silly to believe that this bet doesn't narrow MP's range somewhat, but unless BB believes that MP is making this bet with JJ+, AJ and less than half of his AK's, pot odds still dictate a call.

River is the K♠. BB checks, and MP, overbetting the pot slightly goes all in. BB calls.

I think this is the only identifiable mistake. Of MP's made hands, BB has a chopped pot against AJ and nothing else. To give BB credit for running a three street move (possibly including a bluff raise preflop) on either complete air, or an underpair is just too much risk against the odds being offered. At limit poker, where the final bet might be 10% of the pot it would be a different story of course. At this point, the wave function has collapsed sufficiently for us to observe that, unfortunately, the cat is dead.