Math and Intuition

Poker players sometimes distinguish between intuitive and mathematical play. But most interesting strategic decisions have an element of both, because the mathematical analysis depends on estimates that you can make intuitively but not formally. A simple example of this comes when you need to base a decision on the range of hands your opponent might hold. If you know your opponent well, you can make a reasonably informed intuitive guess. If you'd stored your entire lifetime of observations of this player in a database, it's possible you could derive these estimates more formally, using well understood statistical procedures. But it's not usually possible to do that. So, even the most rigorous formal analyses may depend more than anything on good intuition.

A hand that appeared on the Internet newsgroup rec.gambling.poker about a year ago is a nice illustration of this point, and as a bonus presents an excuse to discuss a pretty common no-limit hold'em situation. I'm not exactly a world-class no-limit player, so it's possible I'll miss some important points in the hand analysis. That's OK. The real point I'm trying to make here isn't tied to the details of this particular hand.

The original hand was posted on RGP by Barry Tanenbaum, and comes from a no-limit hold'em ring game with blinds of $1 and $2. To make it more exciting, I'll put you in his place. Holding pocket aces under the gun, you make it $4 to go, and get called by a late-position player and the big blind. The flop comes A-8-6 in three suits. You check and call the late-position player's bet of $12, making the pot $37, now heads up. The turn is a 10 in the fourth suit. Convinced that your opponent will take a free card this time, you decide to bet, with $130 in chips in front of both you and your opponent. The question is: How much should you bet?

To figure this out, we need to know your expectation (EV) as a function of two factors: bet size and the probability your opponent will call (which is itself a function of bet size). Calculating EV in this case is straightforward, if slightly tedious. First we need to estimate the probability of each possible hand your opponent might hold. Then, for each of those hands, we need to estimate the probability of each possible outcome of the cards (the hands you both hit, and so on). Finally, for each of those possibilities, we need to assign a probability to each possible cash outcome.

In Barry's original post, he wrote that he was confident his opponent held one of three hands: 9-8, 8-7, or 7-6. I've simplified by assuming no flush draw is possible. If that's the case, it's easy to calculate the probability of different outcomes on the river, because all three hands would be an underpair with an insight-straight draw (and so would a few other hands, like 10-9, 10-7, and 9-6). If we believe our opponent would play all of these hands identically, we effectively have only one case to worry about. So, first let's figure out what can happen on the river. Of the 44 cards unaccounted for, the following outcomes are possible:

Obviously, the deck is loaded with blanks. Only 36 percent of the time does anyone improve on the river, and only 9 percent of the time does your opponent outdraw you. For the EV calculations, I've assumed you will always pay off his straights, and he will pay you off whenever he makes trips, but only half the time when he makes two pair.

To calculate your EV as a function of your bet size, we can consider the $37 in the pot dead money. So, if your opponent folds, your EV is $37. If your opponent calls, your EV is a weighted sum of the contributions from each possible outcome. The weights are just the probabilities of each outcome. So, for each row in the table above, we can take the probability, multiply it by the EV, and add it all together. I've done the dirty work for you, and it comes to 32.07 + 0.83n, where n is the amount you bet. That's your EV if your opponent calls. Not surprisingly, it's clear that your EV goes up with bet size, and that your best outcome will come from making the largest bet your opponent will call, as long as you bet at least $5.94. When you bet exactly $5.94, it makes no difference to your EV if your opponent folds or calls, it's $37 either way. But if you bet less than that, your opponent can cut into your EV by calling. So, that's the minimum you should bet.

Since your opponent may call some bets and not others, your EV for a particular bet size is also a function of p, the probability your opponent will call. Doing a little arithmetic yields your overall EV as a function of n and p: 0.83np-4.93p+37.

Now we're getting somewhere. The only problem left is to figure out what p is for all possible n's. This is a little trickier. One thing we could do is assume that p is some simple function of n, say (130-n)/130, so that your opponent's probability of calling varies linearly between 0 and 1 as your bet goes from $130 down to $0. In that case, it's easy to find the maximum, which occurs when you bet $68 (for an EV of $62). Does it seem plausible that our opponent would be 50 percent likely to call a $65 bet into a $37 pot with just those hands? Perhaps we should assume your opponent will fold to anything more than $80, and p varies linearly beneath that, at (80-N)/80. In that case, your best move would be to bet $43, for an EV of $51.

It's likely that neither of these is a particularly good description of what size bets your opponent will call. An even better approach might be to use your knowledge of your opponent to guesstimate p for a variety of n's, and see which one really maximizes your EV. That will let you take into account all kinds of special cases, like the possibility that an overbet of the pot will be likely to arouse some suspicion, or that your opponent would never call certain sized bets.

Now we have an answer – $68, $43, or some other number. You can quibble with many of the details, and as soon as we change the setup a bit, you have to redo a lot of the arithmetic. We haven't probed the sensitivity of this analysis, or thought deeply about the cost of being wrong. But clearly, although we've taken an overtly mathematical approach to evaluating a particular poker situation, we've produced an answer that's far from objective. It depends on several unknown values that we've simply guesstimated, based on our subjective knowledge of our opponent, in order to get an answer.

What are the hands your opponent might hold? We assumed three hands, each equally probable, based on knowledge of your opponent. How likely would you and your opponent be to pay off on the river? I've assumed you'll always pay off, and your opponent will do so half the time with two pair and all the time with trips. But perhaps there's a chance you'll pick up a tell, or your opponent will fall in love with two pair despite the apparent danger. What's the probability your opponent will fold to a given bet size? It's unlikely that the relationship is anywhere near as simple as the linear function of bet size we assumed.

Our working assumptions are necessary to solve this problem, and hopefully are reasonable. But they're not based on rigorous analysis or mathematical formulae. They're based on our knowledge of (or guesses about) the player(s) involved. There's no simple formula that will tell us how often your opponent will be able to release two pair or trips on the river. If we had detailed records of this opponent's play, or perfect memory, we might be able to narrow things down a bit. But in practice, that's usually impossible. Ultimately, we have little choice but to use both math and intuition to solve questions about poker strategy.

There's another important way in which mathematical analysis is closely tied to intuition. The answer we came up with above is very specific to this exact situation and our exact assumptions. Change one detail and we have to do all the math over again. Since you don't have the time to work all this out at the table, and it won't come up exactly this way again, why bother? Well, although the exact numbers won't necessarily help you, the process of working it out with different sets of assumptions will give you a much better feel for how to make these decisions intuitively. If you've never worked out an example like this, you might be hopeless at guessing what the right bet size should be. But if you've worked out problems like this many times, in a range of situations, at least you'll have a fighting chance. In other words, while good intuition supports your mathematical analysis, the converse is equally true.