Multitable Tournaments as a Series of N-ups

By Tony Guerrera

The Poker Helper article, When to Risk All Your Chips Early in a Tournament, models poker tournaments as a series of double-ups and suggests that the minimum edge you should be willing to take with your tournament life on the line is lower than what has been advocated by the pundits. This article generalizes those results by treating tournaments as a series of n-ups (for example, a 2-up is a double-up, a 2.5-up is a two-and-a-half up, and a 3-up is a triple up).

The Theory

According to the work in When to Risk All Your Chips Early in a Tournament, the probability of winning a tournament is given by , and the probability of finishing in ith place, where ith place is any place other than first place, is iterative formula, . In these equations, p is the probability of you winning each double-up, and E is the number of tournament entrants (including yourself). For the case of n-ups, these iterative formulas can be generalized:

Using the results of these equations, you can find the monetary expectation value (mEV) for any tournament by summing P(ith)(Payout for ith) for all values of i (in other words from i = 1 to i = E):

mEV = P(1st)(Payout for 1st) + P(2nd)(Payout for 2nd) + … P(Eth)(Payout for Eth).

If you want to reverse this process, you can start with a target mEV corresponding to your target return on investment (ROI), and then solve for the value of p corresponding to that edge. Solving for the target ROI is a pain to do by hand. You can set up a spreadsheet in a program like Excel, but it’s still a hassle there. Fortunately, I wrote a program to do it for me. Appended to the end of this article are results I obtained for various target ROIs for the payout structure found in many medium buy-in tournaments, high buy-in tournaments, and rebuy tournaments on Poker Stars.

Interpreting the Results

If you go to a site like Official Poker Ranking (www.opranking.com), you can see the ROIs of every player who’s ever played a multitable tournament in the online poker rooms that OP Ranking tracks. Look at the results of players who’ve logged at least a few hundred tournaments, and you can see that an ROI above 100% puts you in a very elite class.

The typical chip preservation argument says that skilled players should preserve their chips in marginal situations, so they can outplay their opponents in later situations when they have a bigger edge. However, sitting and waiting for too long results in losing a chunk of your stack to the blinds; by the time you get your prime opportunity, you can’t win as many chips. The n-up model shows that 100% ROI players enjoy an edge in the 57%-60% range per double up for smaller tournaments (250 entrants or fewer). In larger tournaments (300 - 1,500 entrants), their edge per double up is 55%-56%. This isn’t what most people think of when they say “big edge,” but remember that since you’re always playing against hand distributions, it’s tough to be more than a 65% favorite against a single opponent.

All right, what about when some dead chips are in the pot? Or what about situations when 2 or maybe even 3 players are all-in. The basic guideline is that as the number of chips increases, the minimum winning percentage decreases. After all, you need fewer triples-ups than double-ups to win a tournament. The additional chips available in the pot make a big difference: the minimum winning percentage for a 100% ROI player begins to dip below 50% for 2.5-ups. And when it comes to triple ups, a 100% ROI player essentially progresses though a series of triple-up, where he wins each one about 40% of the time.

This is an extremely important point about tournament poker. Many players believe that they shouldn’t even think about putting their tournament life on the line when they aren’t at least 50% to win. Those of you cash game players out there know that poker is really about pot odds. And though chip EV and monetary EV aren’t directly proportional in tournaments that pay multiple spots, additional chips in the pot reduce the minimum winning percentage required to call-even to below 50%.

If you’re taking barely positive chip EV propositions for your stack, you’re generally making a mistake. But if you’re not taking any calculated risks to accumulate chips, then you’re probably playing too conservatively. From my experience, playing too conservatively in multitable tournaments results in consistent top third finishes where you either bubble or finish just barely in the money. And with the top-heavy payout structures of most multitable tournaments, you won’t be making much money; overly cautious tournament players are usually losing tournament players.

Beyond the Model

With all this analysis, we need to acknowledge that the n-up model is nothing more than a mathematical model. A mathematical model is only as good as the assumptions that go into it, so with that being said, the big flaw in the n-up model is that tournaments don’t play as a series of n-ups. Highly skilled players tend to play a fair amount of small ball, recognizing opportunities to win pots uncontested, squeezing value out of marginal hands, and trapping unsuspecting opponents with monsters for massive quantities of chips. With this in mind, the results of this model should be adjusted such that the minimum acceptable P(win) for a given ROI is actually higher than what the model suggests.

A precise quantitative adjustment is currently beyond my grasp. But even though I’m the “By The Numbers” guy, I see no harm in some educated qualitative analysis. Forsaking marginal edges in anticipation of bigger future edges is important, but if you’re playing at a fullhanded table, you’re not going to get an overwhelming number of opportunities to assert your superior playing skills. Without good cards, you can only run so many successful bluffs. When you finally get good cards, your opponents aren’t guaranteed to pay you off. With that in mind, my intuition is that the actual edges you should take should be on the order of 2%-3% higher than those suggested by the n-up model.

Tony Guerrera is the author of Killer Poker By The Numbers and co-author of Killer Poker Shorthanded (with John Vorhaus)

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