Combining Math and Psychology

In a recent issue of Card Player (Vol. 19/No. 10), I was interested to see Tom McEvoy write - under the title, "Mike Caro is Right!" - that in poker, math is meaningless and psychology is paramount. However, he did go on to temper this rather dogmatic assertion, implying that this applied mainly to tournament poker rather than cash-game play. I am sure this is basically right; understanding how your opponents play and how they perceive your play is certainly more important than knowing the specific math of a situation. However, there are often circumstances in which a good knowledge of the math can act as a useful aid in the consideration of the psychological factors.

For example, in issue No. 3 this year, I wrote about a method that can be used to help you make decisions when you are not sure where you stand in a hand. In the particular example I gave, a blind-stealer open-raised from the button and you three-bet with A-Q. The big blind folded, the button called, and the flop featured random low cards. Your flop bet then got raised, you called, and the turn was a blank. You checked and he bet. There were now six and a half big bets in the pot, and - perhaps surprisingly - it turned out that if you thought there was a 20 percent chance that the button was either bluffing completely or pushing a draw, it was profitable to call down. In fact, 20 percent was probably a little generous; you can probably call down favorably if you put his bluffing/semibluffing potential as low as 15 percent. So, even if you suspect that you will be ahead here only one time in six, it is worth calling down.

Now that you have the math knowledge, you have a better idea of how to approach the psychological aspects of the situation. If you consider solely the psychological factors, you will only ask yourself questions such as: Does this player bluff in these situations? Is he doing this because he believes I can make a laydown here? Does this player appear to be a total rock? However, once you have a handle on the math of the situation, you can ask yourself a more precise question, for example: Based on what I know of my opponent, is there one chance in six that he currently has a worse hand than I do?

Of course, even if you ask yourself this question and provide a decent answer, this is not necessarily going to lead to your playing the hand in optimum fashion. For example, you may decide to call down and your opponent ends up showing you a weak one-pair hand. However, had you fired back a check-raise on the turn, this might have proved to be a sufficiently convincing play for your opponent to lay down his modest holding, and you would have stolen the pot. Now, we are back in psychologyland, and the math has indeed become meaningless.

Let's look at another example. You are playing shorthanded and are in the big blind with the 9 6. Everyone folds to the small blind, who opens with a raise. You have a reasonable line on the small blind, and he is not a compulsive stealer in this situation. You have seen him fold here and also just call. You also have seen him occasionally decline to complete from the blinds even when offered pot odds of around 6-to-1 or 7-to-1. Thus, you strongly suspect that he has a decent hand here. Nevertheless, you have position with good pot odds, and you call.

The flop brings good and bad news. It is A 7 4, giving you a flush draw but also giving your opponent a big hand if he holds an ace. Considering what you know of your opponent's tendencies, you think this is quite possible. Thus, it is unlikely that you will be able to use the strength of your draw to push him off a better hand.

Naturally, he bets, bringing the pot to two and a half big bets, and it's up to you. Anyone who knows anything about limit hold'em should now instinctively be thinking: "Raise - as a semibluff or as a prelude to taking a free card on the turn." Raising is tempting, and certainly comes into consideration. However, if we then get reraised, we will end up regretting our play, as we will be paying more than necessary for our draw. So, what should we do?

This is an ideal situation for blending math and psychology. First of all, we need to do some math. When we get some answers from the math, we can then apply them to the psychology of the situation, and will have maximum information for our decision. In the following analysis, we are going to have to make some assumptions so that the calculations do not become unwieldy. For example, we may occasionally make our flush and lose, or we may occasionally not make our flush but win. Let's assume these possibilities balance out. Let's also assume that since our opponent is solid and almost certainly has a decent hand, it will not be possible to get him to fold and we will have to show down the best hand to win.

1. Passive Play
First let's assume we play the hand passively. We will call the flop and then call the turn, and just hope we hit our draw. We know that a flush draw is basically 2-to-1 to hit with two cards to come. If we miss, it will cost us one and a half big bets. If we hit, we are going to win two and a half big bets plus whatever we can pick up on the turn and river. If this includes raising and getting paid off, it will climb to five and a half big bets. However, sometimes the third flush card will scare our opponent and we may pick up only four and a half big bets; let's say five big bets as a compromise. Thus, playing the hand out three times means that we lose one and a half big bets twice and win five big bets once. We show a profit of two big bets, or 0.66 overall.

2. Successful free-card raise
This time, we raise on the flop and our opponent cooperates by calling and then checking the turn. We now will bet again only if we hit our hand. Thus, if we miss, we will lose just one big bet. If we hit, we will pick up three big bets (there were two and a half after our opponent's flop bet, and his call of our raise has brought it up to three plus whatever we get on the turn and river. If we are lucky enough to hit on the turn, this could be five big bets, but if we hit on the river, our opponent may well be suspicious of the third flush card (especially after our turn check) and check-call, thus limiting us to four big bets. Overall four and a half big bets seems like a reasonable payoff. With this scenario, playing the hand out three times means that we lose one big bet twice and win four and a half big bets once. We show a profit of two and a half big bets, or 0.83 overall.

3. Unsuccessful free-card raise
This time, we raise on the flop but our opponent refuses to crawl into his shell and instead three-bets. Now we revert to our passive strategy, but we have paid more than the odds for our draw. When we miss, our draw we will drop two and a half big bets. On those occasions when we get lucky, we will pick up three and a half big bets plus whatever we get on the turn and river. Let's say that, as in scenario No. 1, this turns out to be another two and a half big bets, bringing the reward to six big bets. Now, playing the hand out three times means that we lose two and a half big bets twice and pick up six big bets once - a profit of one big bet, or just 0.33 overall.

In summary, we can see that the successful free-card raise yields the best outcome - gaining us 0.17 of a big bet over passive play. However, when we get three-bet, this ends up costing us 0.33 of a big bet - basically twice the gain from the successful free-card play. Now we can make an informed decision. Our free-card raise needs to work two times out of three for it to give good value. If it turns out that our opponent is three-betting us half the time, we are better off being passive. Now that we know what the numbers are, we can again ask a precise question: Based on what I know of my opponent, is there one chance in three that he will play back at me?

But - I hear you asking - isn't all of this far too complicated to work out at the table, especially in an online game in which you have maybe just 15 seconds to make your play? Yes, of course it is. But you don't need to do it at the table; you can do it as homework. If you play a lot of shorthanded poker, this and similar situations will arise again and again. If you spend just an hour or so fiddling around with pot sizes and different numbers of outs, you can develop a very good feel for the probabilities in various situations. You then can combine this data with knowledge of your opponents and make decisions based on a powerful blend of both psychology and math.

Byron Jacobs is the author of How Good is Your Limit Hold Em? with Jim Brier, and Beginner's Guide to Limit Hold'em. They are available through bookstores and at www.dandbpoker.com. Byron may be contacted at [email protected].