Going Against the Maths
Monday, 9 September 2013
...is sometimes right says Jeff Kimber.
When someone finds out you’re a professional poker player, one of the first conclusions they’ll reach – having already assumed you’re some degen gambler who spends all day in your local bookies – is that “you must be really good at maths”.
There’s no getting away from it – maths plays a big part of poker and, if you think back to your GCSE maths classes or certain ex-girlfriends, I’m sure you can pick some people who just never got on with maths. They needed to use all fingers and toes to count and would never be able to make poker players, even if they possessed other skills that would come in handy.
When the cabbie, or whoever it is, asks if poker’s all maths – or if it’s all about being able to bluff while holding your poker face, or all about picking up tells, or all about knowing when to gamble, play tight, or whatever else – the truth is it’s a massive combination of all of these, plus a lot more, but you can’t be over-reliant on any.
Most of the skills aren’t quantifiable. You can think a guy’s bluffing – his body language tells you so and his bet-sizing doesn’t make sense – but he’s playing really tight and you’ve played this guy before, and he always has it.
You’ve shown down some really bad hands of late, though, and you know he’s good enough to pick up on that and know you’re playing on the loose side. But you know he’s struggling to pay his rent and, with the bubble approaching, he’s unlikely to take a risk when he could play snug and lock up a payday.
All of these factors will go through your head as you tank and think about your next move, but none of them can be given a reliable value.
You can estimate that, going on the live tells and bet-sizing reads, he’s 75% bluffing. But in reality, if someone told you it was 85% or 65%, you couldn’t really argue your case that you’re “right”.
The same can be done for all the factors listed above, and once we’ve put all our guestimates together, we arrive at a conclusion, and either call, raise or fold.
Because there are still some unknowns in our equation – namely our opponent’s cards – we have to do a little guessing to reach a mathematical answer. Because online players have played hundreds of thousands of hands, their sample size gets bigger and they get closer at establishing a true range for players, which means they can do the maths and come up with a much more quantifiable answer.
The question is, is it ever correct to ignore the maths and make what is mathematically a “bad” fold to maintain your tournament life? I would argue the answer, under the right circumstances, is yes.
A friend of mine was deep in the Sunday Million. Nineteen left and he was within spitting distance of the final table, a potential chop or the glory of a quarter of a million payday for first.
He rarely plays online – fewer than 50 tournaments a year – and the Million is the only one he plays with this kind of buy-in and prize pool. He plays it maybe 10 to 15 times a year. The average stack is shallow – maybe 25 or 30 big blinds – but my friend has got short, and is playing six-handed, with blinds coming round quickly. He has nine big blinds in the small blind, and the action is passed to him. The big blind has around 15bb and has played pretty snug.
Sounds like an automatic all-in, right? Close your eyes and push? Well, yeah it does, but he does look, and he’s got 9-3 off. Still, with under 10 bigs, the blinds eating him quick and a fairly solid player who will be crippled if he calls and loses in the big, he ships all-in.
We’re taught not to be results-orientated, but it’s hard not to be when it’s your deepest ever run in the biggest comp of the week and you’re in with a chance of your biggest ever score. The big blind called with A-9 and he was out of there, close but no cigar.
Ever since, that move has been eating away at him. Should he have pushed? He asked me the next day what I would have done. I said I think it’s a shove, but whether I would have actually done it, I doubt. I think I’d probably pass and look for a better spot, hopefully in the next four hands before I have to pay another big blind.
He said, "Surely, if you think it’s a shove, it’s a shove," and mathematically he’s correct. If I played this tournament a hundred times, or a thousand times, and got to the last three tables, under 10 bigs, over the long term, I think it would be profitable to shove.
However, I’m realistic enough to know I probably won’t reach the last three tables five times in my life, so how can I base my decision on a long term I know I’ll never reach.
Not happy with my answer, he went to another mate, an online MTT wizard, and asked him. He said he thought it would be a shove, but would work through it with him using some software.
First, they assigned the big blind a calling range – all pocket pairs, all suited aces, A-5 off +, K-8 suited +, K-9 off +, Q-9 suited +, Q-T off + and J-T suited. He then fed that range into Odds Oracle and worked out whether shoving 9-3 was a profitable move and came up with the following results:
We have 29.72% against the big blind’s calling range, which obviously has 70.28%. However, if his calling range is correct he will only be calling 26.67% of the time (as he gets dealt worse than this range 73.33% of the time).
We can therefore work out if we make a profit on average. When he folds, we make 2.5bb profit, so on average 0.7333 x 2.5 = +1.83bb
When he calls, we win 29.72% of the 19bb pot, so 0.2972 x 19 = 5.65bb, a loss of (8.5-5.65) 2.85bb. Therefore the 26.67% of the time he calls we lose 2.85bb, or (0.2667 x 2.85), an average loss of 0.76bb. We can therefore mathematically prove, assuming we have the correct range, that we win 1.07bb every time we make this move (1.83bb when he folds minus the 0.76bb loss when he calls. Of course that is the long hand; Odds Oracle tells us in seconds that the 9-3o shover nets 1.0767bb on average).
Job done, said the online pro. It’s a shove, the maths say so. Next case.
The problem with basing every decision on an average long-term mathematical profit is that we won’t play enough in this situation to realise the profit, and the short term variance might kill us.
How many times have you heard poker players discussing Deal or No Deal and heard them ask, “How could they take that?!” Poker players are not the only people who can work out that when a contestant has 1p and £100,000 left, the average is £50,000.
So when the Banker offers £33,000, the outraged poker players shout at the screen, ‘You can’t deal there, he’s robbing you. I’d take £50,000 or open the box, it’s a terrible deal’.
The problem with that attitude is half the time you’ll win 1p, the other half £100,000 (we all know the Banker doesn’t negotiate!), and if you win 1p this time, there isn’t a next time for you to realise the long-term average of £50,000.
Returning to our Sunday Million scenario, the conclusion that it’s a fold, even though the maths says it’s a long-term profitable shove, is made clearer by considering ICM.
The Independent Chip Model gives a $ value to your tournament chip stack compared to the overall prize pool, and it’s here that we gain support with our fold despite the mathematically profitable all-in.
Adding an average of 1bb to our 8.5bb stack (should we fold) is not going to make a great deal of difference to our tournament. We will need to shove again imminently and need to double up to have a chance of a deep run. Getting knocked out will make a great deal of difference, though! We know if we get called we’re in a world of hurt, so the risk of winning 2.5bb is not worth the risk of losing our tournament life.
My initial instincts were that shoving 9-3o will probably be correct mathematically, but that in this tournament scenario it wasn’t worthwhile, and this is backed up by the maths and the ICM.
Going back to our make-up of a poker player, two often overlooked qualities are instinct and experience, which go hand in hand – as you play more you’ve seen a lot more scenarios and you hone your natural instincts.
Thankfully, mine seem to still be pretty sharp. I just need to prove it by getting to the business end of the Sunday Million to put them to some use.