A Poker Logarithm

Many people play no-limit hold'em in the tournament context. Without a huge outlay in money, they are able to enjoy the excitement of the game. Making the final table and being able to push around large stacks of chips can be a real thrill.

One number I like to keep in the back of my mind when playing in a no-limit tournament is the average stack size for the final table. I like to compare my current stack size to that number in the back of my mind. In no-limit tournaments, it's possible to accumulate chips rapidly. People speak frequently of doubling through, and in a no-limit tournament, it's possible to double through even when the blinds are small relative to stack sizes.

Suppose the final-table players have an average stack size of 8,000. You have 1,000 chips. If you double through three times, you'll have 8,000 chips. If you have 2,000 chips, then doubling through two times takes you up to the average stack size. Notice that 8,000 divided by 1,000 is 8, while 8,000 divided by 2,000 is 4. In addition, note that 8 is 2 raised to the third power, while 4 is 2 raised to the second power. That is, the base 2 logarithm of 8 is 3, and the base 2 logarithm of 4 is 2. Since logarithms need not be integers, this suggests a general index of how much one has to double up to reach final-table average stack size. What we do is divide our current stack size into the final-table average stack size and then take the base 2 logarithm of the quotient. Let's call this base 2 logarithm our doubling index.

Let's suppose your doubling index is 3. By its nature, doubling through implies you have all or most of your chips in the pot so surviving a double-through event is very important. Plus, many of these situations involve heads-up confrontations. If you end up all in and heads-up, you want to be a big favorite. For example, if you have a .9 probability of winning each such confrontation, then the probability of successfully doubling through three times is .729. This means you are going to have average stack size for the final table about three out of four times which is pretty good.

Of course, the preceding is simplistic, but it is worth thinking about because it does have implications about how you should be playing. For example, if it is early in the tournament, the blinds are small relative to your stack size, and your doubling index is around 3, then you can afford to be patient and wait for an opportunity to strike. Patience is important because good doubling opportunities (that is, you are a big favorite to win the pot, and you have one or more players who are willing to go all in or match your all-in bet) are relatively rare. You may not see such an opportunity even once, but it is worth keeping in the back of your mind just how close you are to reaching final-table average stack size. You have to combine this with other parts of the game.

Let me now tell you of a recent situation that happened to me in the Harvest Poker Classic at Casino Regina last November. The tournament was the no-limit hold'em tournament, and we were in the midst of the second level following the break. The rebuy period was over. I had about 3,000 chips with a doubling index of approximately 4.3. The blinds were $50-$100 so I had plenty of chips. The under-the-gun player limped in for $100, as did the player two to his left. They both had stacks a little bigger than mine. The remaining players folded around to me on the button. I peeked at my cards and found K-K.

With powerful hole cards like kings, I'm going to try to double through. I knew the under-the-gun player well and could easily read his hands. The second limper was a stranger, other than the 70 minutes I had been watching him, and I had developed some feeling about his play. Thus, I raised to $500. The two blinds folded; both limpers called.

The flop came 4-4-J; the under-the-gun player bet the minimum of $100. As I said, I could read this player and knew he had a jack with either a nine, 10, queen or king. The latter was not as likely because of my pocket kings. The second limper folded. We were now heads-up. I raised to $1,000 hoping to win the pot right then -- but didn't mind if he called. He thought a little while and called.

The turn card was an eight; he checked. Of the 44 unknown cards (I was assuming I knew his cards to the extent he had a jack plus one of the cards mentioned above), I won the hand if any card other than a jack came on the river. Thus, I had a probability of slightly more than .95 of winning the hand. I gladly took those odds for doubling through and bet the rest of my chips.

He pondered for a while and finally called. When he did so, I turned over my pocket kings and he would not show his hand which was a telling sign of something. Well, we play this game because of the vagaries of chance and when I saw a second jack come on the river, I started to stand up as I knew I was cooked. He slowly turned over his hand and exposed K-J.

So I had read him correctly but, in spite of that, he was the one stacking my chips. I didn't feel badly because I got him exactly where I wanted him; now and then a two-outer is going to bite back. I will bet in the future when I am a big favorite because I know who usually stacks the chips in these situations.