Tuesday, August 3, 2010

Classic hand, classic game theory

Last year the Fitz hosted the so-called Ladbrokes Face Up Holdem World Championship. Myself and the brother were excited about this novel form of poker where the only skills were purely mathematical, and we sat down to work out optimal strategy in advance. We were convinced some others would have it worked out even more, but that didn't seem to be the case. At the start, people were playing appallingly. Most seemed to be sticking to their 'normal game' - doing things like playing suited connectors (despite zero implied odds), not realising that unless it was an allin situation, it didn't matter that two overs were almost 50/50 against a pair (the important odds being after the flop, if the pair can then push the overs out), just going allin with the best hand (not allowing anyone to make a Fundamental Theorem mistake) etc. etc. As the tournament went on though, the optimal strategy seemed to propagate virally: the players who started playing badly now starting to ape what they saw the better players doing.


Anyway, I had a good run just falling short of the final table (card death is unsurmountable in this form of poker, alas). The brother went all the way and claimed the title. The hand that swung it his way happened when he was three-handed against Cecil and bone fide Irish poker legend, Colette Doherty. Sean called a two way all in with 52o when both Colette and Cecil had KQ. This hand was reported in Poker Europa, leading to a bemused reaction among some of Sean's friends who couldn't believe he'd called an all in with 52o when he knew he was up against KQ. In the words of one, "Well done, but you were in with shite. SHITE!"


It was generally believed that Sean made a mistake calling the allin. Actually, it was a brilliant move that only someone with Sean's understanding of the odds could make. Here's how the hand developed.


Cecil, on the button, looks down at KQo, sees that Colette in the small blind has the same hand (KQo), and Sean (the shortest of the three stacks) has 52o.


Cecil moves all in.


Colette, presumably thinking that Sean will fold his garbage hand and not wanting to surrender her SB and half of Sean's BB to someone with the same hand, calls.


Sean instacalls.


Like I said, most people's instinctive reaction was to think Sean had made a mistake. In actual fact, he's the only one of the three players who didn't.


Against one KQ, Sean would be almost a 2 to 1 dog, so it would be a clear mistake to call, as he'd be getting only a bit better than evens on the call. Against two KQ's, he's getting better than 2 to 1 (taking his posted BB into account, and the fact that he's the shortest stack). The even better news is that because his two opponents have each other's cards, he's no longer a 2 to 1 dog. In fact, he's favourite to win the hand at almost 40% to both his opponents 30% each (or 60% if it's just one and the other folds). Once you realise this, it's a no brain call.


So if Sean didn't make a mistake, did either or both of his opponents? Yes and no: it's actually a classic game theory situation where the correct move depends entirely on what your opponents do. The correct play for his two opponents would be for one of them to move all in and the other to fold, thus depriving Sean of the odds to call. But poker's not a team sport, and each individual acts in their own self interest.


Let's look at Cecil first. If he thinks either Colette or Sean will fold to his all in, then his all in is correct. But if he thinks Colette may not fold, then he's making a clear mistake, since it allows Sean to call getting better than 2 to 1 on a 6 to 4 shot (of course, if he thinks Sean's not smart enough to realise this and will fold, then his allin becomes correct again). The more correct play, the one which will never be wrong, is for Cecil simply to fold. This would allow Colette to move all in and Sean would be making a big mistake in calling, so would have to fold. Shorty would get shorter and both big stacks equity would increase.


Now let's look at Colette's call. This is a clear mistake unless Sean folds. By calling, she's improving the odds Sean is getting to call with his garbage well past the point where a call becomes correct. In effect, she's handing the short stack the chance to treble through and get back in the game on a 6 to 4 shot.


As it happened, Sean spiked a 5 to scoop the pot and never looked back.

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