The Numbers Game

This is the second installment of my weekly poker column in the Economic Times, Range Rover.

‘Math is over-rated in poker,’ said a friend the other day. ‘Poker is about psychology, reads, getting inside your opponent’s heads. Math, shmath, pah.’ This is a popular view among many recreational players – but they couldn’t be more wrong. In my view, maths is the foundation of poker, and everything else feeds into it. If you do not master the numbers game, you cannot master poker.

Consider what a decision at a poker table involves. You’re in a hand against an opponent. From the information available to you, you try to put him on a range of hands, and modify that as the hand progresses. Your actions depend on two things: the equity of your hand against his range; and the likelihood of his folding or calling at any stage. Simply put, pot equity (PE) and fold equity (FE). Once you estimate those, it’s just a matter of crunching the numbers to come up with the mathemetically correct decision.

Now, your reads and psychological insights are not irrelevant. On the contrary, they’re among the tools you use to figure out your opponent’s range, and how likely he is to call or fold. In other words, they help you arrive at both your PE and FE in the hand. But having done that, it boils down to the math. Here’s an example.

Stacks are deep, you open with AJcc on the button. The big blind flats. The flop comes KJ2 with two hearts. BB checks, you bet, BB calls. You now put him on a range that includes any king he calls with preflop, any jack, middle pocket pairs like TT and 99, the open-ender with QT and any flush draw he called with pre. The turn is a brick, an offsuit 5. Both of you check. The river is another offsuit 5.  He now bets 75% of pot. What do you do?

You’d expect him to check back here with any jack, TT and 99. Let’s say he value-bets every hand that beats you, most probably top pair. And he bluffs with QT and every plausible missed flush draw. Against this range, we have 33% equity. Since he bet 75% of pot, we’re getting 2.3 to 1 to call here, meaning the call is justified if we have 30% equity. We have 33%, so we call.

But let’s say that you are an astute reader of this particular player, and of the situation. He tends to be passive, this session is almost over, he is about break-even after having been down. In this spot, you estimate he’d bluff with a flush draw or QT just 50% of the time, but would value bet a K every time. The numbers change: against the same range, but with the bluffing layer weighted at 50%, you now have 21% equity. You should fold.

Do you see what happened here? Your psychological insight and player profiling, maybe even a tell of strength you spotted, helped you make the correct play. But it was correct because the numbers said so, and your read merely helped you arrive at the right numbers. At the heart of it was the math.

Another example: A player raises from early position, you flat from the button. The flop is king-high with two hearts. He bets. If you choose to raise, what hands are you raising with here? That depends on both your equity against his range (PE) as well as how often he will fold (FE). If he is a nit who will fold 90% of the time, you can raise with complete air here. If he is a calling machine who doesn’t like folding, your hand needs to be stronger. If your reads help you come up with his folding frequency, math will do the rest.

Normally one puts opponents on ranges, and determines fold equity, based on observation and memory: from their past behaviour, we deduce their present tendencies. Psychology plays a part only at the margins. The great Indian offspinner Erapalli Prasanna once said in a cricketing context: ‘Line is optional. Length is mandatory’. Let me paraphrase that in poker terms: ‘Psychology is optional. Mathematics is mandatory.’

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Last week on Range Rover:

The Bookshop Romeo

The Bookshop Romeo

This is the first instalment of Range Rover, a new weekly column on poker I am writing for the Economic Times.

A young man enters a bookshop. He loves books. And he’s lonely. He spots a gorgeous young lady browsing a book by an author he loves, Milan Kundera. Their eyes meet; she looks away shyly. He decides to seize the day. He walks over to her, but just as he begins speaking—‘Hi, that’s one of my favourite Kundera…’—a hunky young man appears on the other side of this lady, and she squeezes his arm as he apologizes for having made her wait. Then they both turn to our young man. ‘Yes?’ she asks.

‘Erm, I was just saying, that’s my favourite Kundera book.’

They look at him blankly. ‘Who’s Kundera?’ she says.

And at this awkward moment, dear reader, I have a question for you: Did our hero make a mistake?

The answer to that lies in mathematics. And I will try and explain it through poker. Welcome to this first instalment of Range Rover, my weekly column on poker. The column is meant not for the complete layman, but for the hobbyist who knows the basics of the game. My reflections will be about the technical and mental aspects of the game – and also, sometimes,  about life itself. This first piece, with particular relevance to our Bookshop Romeo, is about ranges.

A mistake beginning players often make is of putting their opponent on a particular hand, and then seeing if they’re ahead or behind – instead of putting them on a range of hands, and calculating their equity against that range. For example, say you raise from early position with AKcc. Villain calls from the button. The flop comes Ks7h2h. You make a continuation bet, villain raises. What do you do?

In this spot, you need to figure out what range of hands villain could be doing this with. If he is a super-safe ABC nit who will only dare to raise here with hands that beat you—basically sets, AA and another AK, as no two-pair combo calls preflop—then your equity against his range is around 20%, and you must fold. If he is spewy-aggro and his range includes all flush draws and worse kings like KQ, KJ and KTs, plus some air, then you are around 60% in the hand and should continue. Now, sometimes you will call the spewy player and find that he has 22, but that doesn’t make your call a mistake: you made the right decision, but ran into the top of his range. Similarly, if you fold to the nit and he shows AK, it doesn’t mean you made a mistake there either.  The result of the hand has nothing to do with the correctness of your decision.

A beginning player would have put his opponent on a particular hand, and congratulated or berated himself based on whether he won or lost. But that would be a mistake, and there is a reason poker players are told not to be results-oriented. Your goal in poker should just be to make +ev decisions against your opponent’s ranges, and not think of immediate outcomes. To get the money in as a 60% favourite will make you rich in the long term – but losing four such hands in a row, as does happen, should not lead you to question the inherent correctness of your decisions. To paraphrase Krishna from the Bhagawad Gita, do the right thing, don’t worry about the fruits of your actions.

Let’s go back to our Bookshop Romeo. He is single, and sees a girl he likes. From the range of possible personality types, he narrows her down to potentially compatible ones because she is in a bookshop and holds a Kundera book. Furthermore, considering that the momentary embarrassment of being snubbed is not much of a cost to bear, given the benefits that are possible, he is getting practically infinite odds to make his move. So he does. It ends badly, but it wasn’t a mistake. Indeed, to not approach the girl would have been an error. He lost this hand – but he played it right.