Misapplying Expected Value Calculations

There is a long-standing argument in the poker world between the math guys and the instinct guys. I have always been a proponent of the math side, as every decision made in poker has an underlying mathematical foundation (whether you realize it or not). Of course, I also wrote a book called Texas Hold’em Odds and Probabilities, so I might be biased. Some instinct guys will say that you sometimes have to throw math out the window, but they are missing the point. If your instincts make you feel like your opponent is bluffing, you increase the probability that your opponent is bluffing in the mathematical calculation. If your instincts tell you that your opponent has the nuts, you increase the probability of this in the calculation. Many players don’t know that they are using math, but their instincts and experience lead them close to what the mathematical calculation would conclude.

Having said this, I want to discuss a particular mathematical play that I have been seeing more and more, and it’s one that many players are misapplying. I’m not sure who started this or when it started, but someone did a mathematical calculation showing that you could profitably shove a wide range of hands when heads up with 20 big blinds. Furthermore, the calculation showed that you could do this with your cards faceup and the play would still be profitable (thereby implying that your opponent is likely to fold a tighter range, making the play even more profitable). The typical scenario is this: It is folded to the small blind, and his stack is 20 big blinds. The word got out, and now you will see him pushing his 20 big blinds all in with quite weak hands.

The first thing to understand is that the profitability of this calculation or benchmark of 20 big blinds depends on a wide variety of factors. How strong is your hand? How big are the antes in comparison to the blinds? In real life, your hand is concealed, so the calculation needs to know the range of hands with which your opponent will call. Some will call loosely, while others will be quite tight. There are other considerations, such as the stack size of your opponent, which might affect his calling range. There are bubble considerations. I don’t want to argue the merits of the calculation in this column, but the benchmark of 20 big blinds is an arbitrary number that’s not applicable in every situation. You could argue that pushing has positive EV [expected value] for ranges even much higher than 20 big blinds in certain cases.

One also needs to understand the difference between chip EV and $EV. Just because, on average, you might increase your stack, the curve of this expectation could very well decrease your tournament equity, or $EV.

The first misapplication of this play is that just because a play has positive EV, it doesn’t mean that you should make it. You should always choose the play that has the most profitable EV. Just to talk extremes, even though pushing all in with A-A and 300 big blinds has positive EV, that doesn’t mean you should do it. For example, if you are up against a tight-passive player, a higher EV play might be to raise three times the big blind. If you are up against a terrible flop player, the better play might be to call and bet the minimum every flop. The point is, you need to evaluate the situation you are in against a particular opponent at a particular moment of a tournament. If you are up against a tough opponent when sitting at a tough table where you have little edge, pushing all in might be the best play. However, if you are at an extremely weak table with a weak opponent in the big blind, there might be better plays.

The second common misapplication of this play is when you have more than 20 big blinds and your opponent has 20 big blinds. In this case, you effectively have a stack of 20 big blinds, and I see many players making the same play based on the “mathematics” of it having positive EV. However, this is a big mistake, as you have to look at the global perspective of how the result will impact your tournament results.

For example, if you have 40 big blinds, picking up the blinds and antes will increase your stack to 42 or 43 big blinds. Those extra blinds won’t have a major impact on the strategies that you can use in future hands. However, when your opponent calls and you lose, your stack is now crippled, as you will have 20 big blinds, which takes away your ability to play pots post-flop. The chips that you lose in a tournament are worth much more than the chips that you gain.

During a World Series of Poker tournament last year, I saw a player push 9-3 offsuit from the small blind. The blinds were 100-200 with no ante. He had 6,000 in chips and his opponent had 2,000 in chips. His opponent had 9-9, and he lost one-third of his stack in an instant. Evaluating this play, there were three end results for this player. Most of the time, he increases his stack from 6,000 to 6,300. When called, most of the time he will lose and end up with a stack of 4,000. Occasionally, his hand will win and he’ll increase his stack from 6,000 to 8,000. The big problem here is that increasing his stack from 6,000 to 6,300 doesn’t increase his tournament equity very much, but going from 6,000 to 4,000 greatly diminishes his tournament EV.

Another situation happened to me recently in an online tournament. My opponent in the small blind had a nice-size stack of 60 big blinds or so. I had 20 big blinds and we were in the money, so the antes had kicked in. He pushed with J-3 offsuit. He will usually go from 60 big blinds to 62 or 63 big blinds, but when I call, he is a big dog and likely to go down to 40 big blinds, greatly impacting his equity in the tournament.

Of course, both of the players in these examples had very weak hands, but the point is that they were not thinking about the end results of their play. Even if they could argue that their play had positive chip EV, the curve of this expectation is such that their tournament EV decreases. The problem is that when they lose, the damage hurts them much, much more than what they gain by simply picking up a few blinds and antes. Even when their opponent calls and they happen to draw out, those chips gained aren’t worth nearly as much as those chips that are lost.

It is great to discover non-exploitable plays, but against exploitable players, there are likely to be better plays. Always make sure that you are choosing the most profitable strategy while keeping an eye on the global picture. 

This article first appeared in Card Player magazine. For more poker strategy, go to CardPlayer.com.