Poker Digest Vol. 3, No. 8, April 7 - 20, 2000
In response to a question from the ubiquitous Bib Ladder, in the previous article we discussed how people come up with the statement there is a 31.1 percent chance of winning with pocket aces against nine random hands played to the end of the hand. We also discussed the margin for error in the figure and concluded we lose nothing by using 31.1 percent as the correct figure.
There was a second part to Bib's question: ``Should a player pay any attention to this number or is it only a mathematical curiosity?'' I maintain it is a meaningful number and following is what I explained to Bib.
Mathematics is teeming with so-called lower bounds and the value .311, the probability of winning with pocket aces with nine opponents who stay all the way to the end, provides a lower bound in the following sense. Suppose you are dealt a pair of pocket aces against nine opponents. If no one folds before the hand is played out, then the outcome is predetermined because all players have whatever hands they have and the cards which will form the board will not change. (The winner is not predetermined in seven-card stud because players folding changes the recipients of subsequent cards.) So at this juncture the probability you will win at the end of the hand is .311 and you will not do worse than this in the long run other than small fluctuations due to the fact chance is involved. So in this sense .311 is a benchmark you will achieve no matter what you do.
There are three possible objectives in playing your pocket aces:
1. You may attempt to increase the percentage of pots you win;
2. You may attempt to increase the return per unit amount you bet; and
3. You may attempt to increase your average profit per hand when dealt pocket aces.
The first objective -- increasing the percentage of pots you win -- is easy to analyze. Let's return to the situation where the outcome is predetermined and no action has taken place. If you are in a situation in which your aces will win at the end of the hand, then any action taken during the hand cannot change that outcome, unless you muck your hand for some reason. On the other hand, by raising preflop some of the predetermined situations in which a poor hand would have won because of a fluky board can be altered. For example, the board may end up with 2-2-5-9-Q and the only hand which beats your aces is some clown with 2-8 offsuit. If a preflop raise induces this player to muck his hand, the predetermined situation in which your aces were going to lose has been altered to a situation in which they now win.
It's easy to see that with pocket aces, action taken during the hand is going to have a much stronger tendency to increase the percentage of pots you win than to decrease the percentage.
As a player you actually have a fairly small range of actions you can take during the course of the hand. You can check, call, initiate a bet, raise, check-raise or fold. Raising is a preflop action which tends to drive out weaker hands. Every hand you drive out before the flop is one less hand which can beat your aces. Thus, preflop raising tends to increase the percentage of hands you win. Just calling preflop actually has a tendency to encourage weaker holdings to remain which has a tendency to decrease the percentage of hands you win. Just calling preflop is done now and then by some players for strategic reasons but it is uncommon. Folding pocket aces preflop is extremely rare but occasionally done in a tournament setting.
Once three or more cards are face up on board, you lose leverage for trying to induce players to fold holdings which might beat your pocket aces. This is highly dependent on the texture of the game. There are many low limit games in which a player who has gotten any kind of help from the flop will be there until the end. Many times aggressive action on your part is still the best way to drive out hands which might beat you. Near the end of the hand perhaps the most effective option for increasing the percentage of pots you win is being able to muck your pocket aces when you are certain you are beat.
Increasing the percentage of pots you win with pocket aces is not the most important objective. This objective plays a role but it should not be uppermost in your mind. What should be uppermost in your mind with pocket aces is to maximize the average profit per hand. This isn't the same as trying to maximize the return per unit bet. The question then becomes how can you maximize the average profit?
Unfortunately, the latter question is not easily answered. Let's think about some of the factors involved in making this a difficult question.
First, can we establish any kind of rough benchmark for profit? Let's consider the unrealistic scenario of nine other players staying all the way to the end, where there is no raising, you bet every round, and all players call. This means the pot will have 60 bets in it -- including your six bets (we are assuming the game is x-2x hold'em and ignoring the rake). You will win roughly thirty percent of the time (this slight reduction partly covers the rake) which works out to an average profit per hand of 12 bets. (Note that your expected return for every bet you make is three bets.)
If you make a preflop raise, what happens next depends heavily on the texture of the game. In some games it typically may result in only one or two players calling the raise, while in other games it may set off a raising war with the betting being capped and seven or eight players calling all raises. In the former case, there will be something like seven bets in the pot (assuming the big blind is not one of the callers) before the flop.
In a game like that, where players are more selective with the hands they choose to play, it's difficult to assign a probability to your aces winning the hand because you're not facing two random hands. Thus, it's difficult to work out an expected profit.
In the second case, there will be something like 28 bets in the pot and the hands you are facing are closer to being seven random hands. One can assign a probability to the aces holding up using the figure for aces against seven random hands but there is some error already creeping in. The point is that action taken during the course of the hand damages the randomness, sometimes quite severely, making analysis difficult.
In the last Poker Digest I mentioned the random sampling done to determine the probability of pocket aces winning against a fixed number of players who stay to the end doesn't deserve to be called simulation.
However, the above discussion opens the door to a problem begging for simulation. Namely, create some player profiles and then play pocket aces in a variety of tables to see what playing strategy produces optimal profit.
I'm not aware of anyone who has done the preceding simulation, but it wouldn't surprise me if someone has. I also wouldn't be surprised if optimal play, in games with certain textures, differs from common belief of how pocket aces should be played. Our intuition sometimes leads us astray.
Without such information all a player can do is attempt to follow what seems to make sense and it's a subject worthy of thought.