By Tony Guerrera

When it comes to tournament poker, the traditional advice given by most authors is to play very conservatively in the early stages, avoiding any confrontations that might put your entire stack at risk. The rationale behind this advice is that if you are highly skilled, you can pass off marginal edges in the present because of bigger edges that you’ll have in the future. Understanding this concept is extremely important if you are to be a successful tournament player. However, some players take this concept to extremes that are ultimately detrimental to their results.

Tournament poker is about balancing the ideas of chip preservation and chip accumulation. On one hand, as you survive longer in a tournament, your payout goes up. There’s an intrinsic value to simply having a seat. On the other hand, you can’t simply sit and play no hands. You need to get chips to keep up with the increasing blinds. Players who constantly pass off opportunities to accumulate chips in anticipation of better future opportunities cost themselves prize pool equity in the long run.

The question we really need to answer is this: what’s the minimum edge that you should accept when risking all your chips in the early stages of a tournament? For example, if your are something like a 57/43 favorite, should you risk all your chips or fold in anticipation of a better opportunity? To answer this question, we need some way of quantifying your edge in a tournament.

One way of doing this is to model your path through a tournament as a series of double ups. Suppose you are in a 16 player tournament, in which the winner takes all. One way of winning this tournament is to double up 4 times. The probability that you will win the tournament is given by , where D is the probability that you’ll win an all-in confrontation and double-up, and n is the number of double ups you need to win. Assuming that you and every one else in the tournament is equally skilled, the probability of you doubling up when you are all-in is .5. Therefore, , which is what we’d expect. If you and all your competitors are equally skilled, then you all should have a probability of winning.

If you have an edge in the tournament, your probability of winning each double up should be greater than .5. How much bigger than .5 is it? The only way of coming close to accurately answering this question is to use data from your past tournaments. Suppose you’ve played 100 such tournaments, and you’ve found that your EV is 2 buy-ins (in other words, your average profit is 1 buy-in). If your EV is 2 buy-ins, then your probability of winning is instead of . You still need 4 double ups to win the tournament, meaning that…

If your edge in this tournament is 1 buy-in, then the probability of you winning each all-in is about .595.

It may seem silly to model a tournament as a series of double-ups because this isn’t necessarily how the actual dynamics of tournament poker play out. However, this calculation does give us some type of quantitative estimate of the type of edge we should be searching for when confronted for a decision for all of our chips early in a tournament. In the example above, this player should risk all of his chips for a double up if his probability of winning is greater than .595, and he should fold, waiting for a better opportunity, if it is less then .595.

What if you are in a winner take-all tournament consisting of 128 players, and your EV is still 2 buy-ins? The math below gives us your value of D in this tournament:

Notice that D is lower in this case. In general, as the field size increases, the risk you should be willing to take also increases. The table below gives values of D for winner-take-all tournaments as a function of the number of entrants, given that your EV is always 2 buy-ins.

So far, this analysis has been of tournaments that are winner-take-all. Most tournaments are not winner take all, which means that we need to generalize this model somehow. Our first step towards generalizing this model is to discuss what your EV in a tournament actually is. At any given point in a tournament, you have a certain probability of finishing in each place, and finishing in each place has a corresponding payout. Your monetary EV, the quantity you wish to maximize when playing in a tournament, is given by the following expression:

In a tournament consisting of N players, we know that , where n is the number of double ups required to win a tournament of N players. To solve for n, we acknowledge that the definitions of n and N make the following expression true: . To solve this equation, take the base 2 logarithm of both sides (if this is too much math for you, don’t worry and just skip to the last paragraph of this article). This means that and .

P(2nd) equals the probability of doubling up to half the chips in the tournament minus the probability of winning the tournament. P(3rd) is the probability of doubling up to a third of the chips in the tournament minus the probabilities of finishing in second and first. In general, the probability of finishing in ith place is the probability of doubling up to a stack of minus the sums of the probabilities of finishing in a higher place. The number of double ups required to get to a stack of is derived below:

Therefore, the probability of doubling up to a stack of is .

In general, the probability of finishing in ith place is given by the following equation:

The only exception to this equation is when i = 1, in which case .

To figure out your value of D for a tournament, you need to solve the following equation for D:

In this equation, M(i) represents the payout of ith place in terms of buy-ins. For example, if 1st place in a $100 tournament is $2,000, then M(1) = 20.

We have the general theory in place, but unfortunately, it’s tough to calculate D using this formula. Because payout structures vary from tournament to tournament, there’s no really good way to make calculations on the fly. Nonetheless, this formula is helpful in that we can use a spreadsheet to apply it to a wide range of tournaments to get estimates of D that we can then take with us when we are actually playing.

Looking at a few different tournaments, with the number of entrants ranging from 100 to 500, I found that D ranged from .58 in the smaller tournaments to .56 in the bigger tournaments if your EV is 2 buy-ins. Of course, your particular value of D should be found by setting up a spreadsheet of your own. Furthermore, this whole model of progressing through tournaments by always doubling up isn’t necessarily perfect–it simply serves as an approximation to be used in an effort to do some rigorous analysis. However, this analysis suggests that you should be a bit more liberal in risking your chips early in tournaments than what is traditionally advocated. In the words of Amir Vahedi: “In order to live, you must be willing to die.�

Tony Guerrera is the author of Killer Poker by the Numbers

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