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Poker Tournament Satellite Value and Risk of Ruin

 
 
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Poker Tournament Satellites: Why Skillful Satellite Play Cuts Your Bankroll Requirements More Than You Probably Think

By Arnold Snyder and Math Boy
(From Blackjack Forum , Vol. XXVI #1, Winter 2007)
© Blackjack Forum Online 2007

This article will look primarily at how you should estimate the dollar value of a poker tournament satellite. It will look at the factors that make a satellite a good investment, and will discuss how skill at satellites can lower the bankroll you need to enter bigger, more costly poker tournaments, or lower your risk of ruin (RoR) for any given bankroll. We’re going to focus on specific satellite strategies in a later article. In this article, we’ll look at the dollar value of satellites to a player who already has the strategic skills necessary to beat satellites.

First, a definition: A satellite is a tournament that does not award money to the winner(s), but instead awards an entry to a tournament that has a bigger buy-in cost. In some satellites, a small amount of money is awarded in addition to a seat or seats into a bigger event, but this money is a secondary prize, awarded either to cover the winners’ travel expenses or to “round out” the prize pool.

It is the seat in the bigger tournament that is the main prize. There are some satellite pros who play satellites primarily for money, not seats into a bigger event. Many satellites, especially for big multi-tournament events, award tournament chips rather than seats in specific events, and these chips can be sold to other players who are buying into the big events. Our primary focus in this article, however, will be on players who want to use satellites to lower their entry costs into main events.

A single table satellite will usually start with ten players, and will typically award one winner a seat in a tournament that has a buy-in of about ten times the cost of the satellite. Some single-table satellites award seats to the top two finishers. A “super satellite” is a multi-table satellite that will award multiple seats into a major tournament, with the exact number dependent on the number of entrants in the satellite.

The “House Edge” on Poker Tournament Satellites

The first thing we need to consider when analyzing a satellite’s value is the house edge. By this, I mean the percentage of the total cost of the satellite that the players are paying to the house for hosting the satellite. Figuring out the house commission is pretty straightforward. We need only two factors: the total combined dollar cost to all of the entrants, and the total dollar value of the satellite prize pool.

For example, a popular World Poker Tour (WPT) single-table satellite has a buy-in of $125. One of the ten entrants will win two $500 tournament chips plus $120 in cash. The other nine entrants receive nothing. For big multi-event tournaments like the WPT or the World Series of Poker (WSOP), it is not uncommon for the hosting casino to run satellites that award tournament chips that may be used in bigger events, instead of seats into a specific event.

Although the series of tournaments may conclude with a “main event,” usually the event with the highest buy-in, there will be many events with smaller buy-ins that precede the main event. A player who wins two $500 tournament chips may use those chips to enter two $500 events, a single $1000 event, or as partial payment into an event that has a buy-in greater than $1000.

In this particular WPT satellite example, the house collects $125 x 10 = $1250, while paying out $500 + $500 (two chips) + $120 (cash) = $1120. So, the house profits $1250 - $1120 = $130 every time they run this satellite. The house edge from the players’ perspective is: $130 / $1250 = 10.40%. Which is to say, the house keeps 10.4% of all the money they collect from the satellite entrants.

At the 2006 Mirage Poker Showdown, a WPT series of tournaments with a $10K main event, there were seven different single-table satellite formats. All were ten-player/one-winner formats. The least expensive had a $60 buy-in, with the winner receiving a single $500 chip plus $50 cash. The most expensive had a $1060 buy-in, with the winner receiving twenty $500 chips (a total of $10K in chips) plus $340 in cash. The chart below shows the buy-ins of these seven satellites with their payouts, the house commission (in $), and the house edge (in %).

Mirage Poker Showdown (WPT)
Single Table Satellites, one winner format
Buy-in Payout in Chips Payout in Cash Total Payout House Commission $ House%
60 500 50 550 50 8.33%
125 1000 120 1120 130 10.40%
175 1500 120 1620 130 7.43%
225 2000 120 2120 130 5.78%
275 2500 120 2620 130 4.73%
810 7500 340 7840 260 3.21%
1060 10000 340 10340 260 2.45%

One obvious trend is that, in general, the greater the buy-in cost, the lower the house edge. The only exception among this group of satellites is that the house edge on the least expensive ($60 buy-in) satellite is smaller than the house edge on the second cheapest ($125 buy-in) satellite.

Poker Tournament Satellite Value for the “Average” Player

As a satellite player, what does the house edge mean to you? Consider your result in these satellites if you play with an “average” level of skill that gives you neither an advantage nor a disadvantage against the field. Some may be better players than you, and some worse, but in the long run, you will average one satellite win for every ten satellites you play.

This chart shows the value (in $ and %) of each of these seven WPT satellites for an “average” player:

Player Wins One Out of Ten Satellites Entered
Buy-inDollar ValuePlayer Adv. in %
60 -5.00 -8.33%
125 -13.00 -10.40%
175 -13.00 -7.43%
225 -13.00 -5.78%
275 -13.00 -4.73%
810 -26.00 -3.21%
1060 -26.00 -2.45%

Let’s first note that in all cases, the dollar value is negative. “Average” players will lose money in satellites over the long run. Computing the dollar value is simple. Consider the $60 satellite. If you play ten of these satellites, you will invest $60 x 10 = $600 in these ten plays. If you win one, you get paid $500 (chip) + $50 (cash) = $550 in dollar value. Collecting $550 for every $600 you invest is a loss of $50 for every ten satellites you play, which is an average loss of $5 per satellite. So, the dollar value of this satellite to the average player is -$5.00.

Likewise, the “Player Advantage in %” is negative for the average player in the other satellites, as we would expect. Again, the computation is simple. If I lose $5 for every $60 I invest, then my result (in percent) is: -5 / 60 = -0.0833, which is -8.33%. You might also note that if you look at the prior chart which shows the “house edge” on these satellites, the “player advantage” for an “average” player is always the negative of the house edge.

From the professional players’ perspective, this means that to play satellites with average skill is a waste of money. In the long run, it will cost the average player more to enter tournaments via satellite than it would cost him to simply buy-in to the big events directly.

A player who is short on funds might argue that he would never pay $10K to enter a major tournament, but that he is willing to gamble $1060 on a long shot to get into such an event. In fact, this is a good argument if you accept the fact that you are gambling on a negative expectation game. One of the reasons that tournaments have become so valuable to professional players is that so many amateurs are willing to gamble at a disadvantage on satellite entries.

Satellite Value for the “Better-than-Average” Player

Now let’s look at how satellite skill affects the value of satellites to the player. Let’s say you can win one out of every nine of these ten-player satellites that you enter. How would this affect both the dollar value and your advantage (in %) in these same seven WPT satellites?

Player Wins One Out of Nine Satellites Entered
Buy-inDollar ValuePlayer Adv. in %
60 1.11 1.85%
125 -0.56 -0.44%
175 5.00 2.86%
225 10.56 4.69%
275 16.11 5.86%
810 61.11 7.54%
1060 88.89 8.39%

Big difference. The cheapest satellites are still not worth the trouble, due to the high house edge. Few players at this modest level of satellite skill should be interested in entering the $60 satellite, as the $1.11 value is a pretty low payout for a lot of work. And the $125 satellite (which you may recall has a higher house edge) is still a negative expectation gamble. The player who can win one in nine of these has a notably superior result to the “average” player who can win only one in ten, as the more skillful player will be losing only 56 cents per satellite played, as opposed to losing $13. Still, in the long run, the player who expects to win just one in nine of the $125 satellites would be paying less to enter the bigger tournaments if he skipped the satellites and just paid the full buy-in price of the event.

With the $1060 satellite, the one-in-nine player advantage is 8.39%, which is a dollar value of $88.89. For many players, this modest level of satellite skill might make these satellites worth the effort.

The Satellite Professionals

As your satellite skill increases, the value of playing satellites goes up dramatically. A more talented satellite pro can do quite well in ten-player/one-winner satellites if he can win just one of every eight satellites he enters. Here’s how he would do in these seven WPT satellites with this level of skill:

Player Wins One Out of Eight Satellites Entered
Buy-inDollar ValuePlayer Adv. in %
60 8.75 14.58%
125 15.00 12.00%
175 27.50 15.71%
225 40.00 17.78%
275 52.50 19.09%
810 170.00 20.99%
1060 232.50 21.93%

At this skill level, there may be sufficient dollar value to warrant playing these satellites at any buy-in level. With the smaller buy-in satellites, you must decide if the expected dollar return is sufficient to spend, on average, about an hour of your time in satellite play. Many of us wouldn’t work for $8.75 an hour, which is the dollar value of the $60 satellite for the one-win-in-eight player.

But we must also consider that for every satellite we play, we gain more satellite experience, and this should translate sooner or later to greater satellite skill. Like all other forms of poker, you can’t increase your satellite skills without playing them. And this becomes more important as the dollar value increases with the satellite cost. For example, with a dollar value of $232.50, the $1060 satellite will gain you entry into a $10K event, on average, for a cost of just over $8K. That’s a substantial discount.

The Top-of-the-Line Poker Tournament Satellite Pros

The top satellite pros win, on average, about one out of every six to seven ten-player/one-winner satellites they enter. They accomplish this with a combination of skill at fast-play strategies, skill at short-handed play, and skill at choosing weak fields of competitors. No pro wants to enter a satellite and find himself facing a table full of other pros. The value of satellites to a pro is as much a function of his competitors’ lack of skill as it is of his own skill.

Let’s compare the dollar values of these seven WPT single-table satellites for players who expect to win one of every ten, nine, eight, seven, six, and five satellites they play:

Dollar Value per Satellite, if Player Wins Once per:
Buy-in 10 9 8 7 6 5
60 -5.00 1.11 8.75 18.57 31.67 50.00
125 -13.00 -0.56 15.00 35.00 61.67 99.00
175 -13.00 5.00 27.50 56.43 95.00 149.00
225 -13.00 10.56 40.00 77.86 128.33 199.00
275 -13.00 16.11 52.50 99.29 161.67 249.00
810 -26.00 61.11 170.00 310.00 496.67 758.00
1060 -26.00 88.89 232.50 417.14 663.33 1008.00

Let’s also look at the various players’ advantages in percent for these frequencies of wins:

Player Advantage (%), if Player Wins Once per:
Buy-in 10 9 8 7 6 5
60 -8.33% 1.85% 14.58% 30.95% 52.78% 83.33%
125 -10.40% -0.44% 12.00% 28.00% 49.33% 79.20%
175 -7.43% 2.86% 15.71% 32.24% 54.29% 85.14%
225 -5.78% 4.69% 17.78% 34.60% 57.04% 88.44%
275 -4.73% 5.86% 19.09% 36.10% 58.79% 90.55%
810 -3.21% 7.54% 20.99% 38.27% 61.32% 93.58%
1060 -2.45% 8.39% 21.93% 39.35% 62.58% 95.09%

The final column in both charts, which shows the player’s expectation if he is skillful enough to win one out of every five of the satellites he plays, is more theoretical than realistic. This may be possible for a skillful satellite player who always manages to face a very weak field, but most satellite players today are not this weak. Many players are aware of the necessity of taking risks in satellites as the field diminishes and the blinds increase.

Nevertheless, we can see from the charts that a satellite player who is skillful enough to win one out of every six or seven of these satellites will have an advantage in the neighborhood of 30% to 60%, and that is a big enough edge to interest any professional gambler.

Other Poker Tournament Satellite Formats

As mentioned at the beginning of this article, not all satellites are single-player one-winner formats. The two-winner format is also quite common. Typically, a player might pay $200 plus the house fee to win one of two seats into a $1K event. With this format, the average player would expect to win two out of every ten satellites entered, as opposed to one in ten. Likewise, a win of two in eight with the two-winner format would be equivalent to winning one in eight with the single-winner format. And the top satellite pros would expect to win two out of every six or seven played in the two-winner format.

The two-winner format is generally advantageous for both the players and the poker room. Satellites played down to two winners finish faster than satellites played down to one winner. This means that more satellites can be played prior to a big event, with twice as many main event entries generated per satellite. The two-winner format also reduces fluctuations for the players.

If you’re good with spreadsheets, you can easily set up a spreadsheet to calculate the dollar return and house/player advantages for the two-winner format based on the buy-in costs and payouts.

Multi-table satellites, often called super-satellites or mega-satellites, are also very common, especially for major events. For example, the WSOP typically has a $1060 super-satellite for the $10K main event. One seat to the main event is awarded for every ten satellite entries. Here’s a chart that shows the dollar values and player advantages based on the frequency of player wins:

WSOP Mega Satellite, $1060 Buy-in,
One Seat Awarded per Ten Entrants
Dollar Value and Player Adv. If Player Wins Once per:
  10 9 8 7 6 5
$ Value -60.00 51.11 190.00 368.57 606.67 940.00
Adv. (%) -5.66% 4.82% 17.92% 34.77% 57.23% 88.68%

Note that the house edge on this event is 5.66%. These multi-table satellites are a good value for a player on a budget who can only afford to enter one satellite for a shot at the main event. And this is especially true if the player is a good tournament player, but not really all that skilled at single-table satellite play. (And there are many players who are skilled at multi-table tournaments who do not fare well in single-table satellites, primarily because of a lack of experience with making the quick adjustments necessary for the speed and short-handed play.)

For a skillful single-table satellite player, however, super satellites have less value than single-table satellites. The higher-priced single-table satellites often have a lower house edge, and they play out much faster. Big multi-table satellites often take many hours to determine the winners, as opposed to the typical 60-90 minutes a single-table satellite lasts. Time is money.

Using Satellites to Lower the Buy-In Costs of Major Poker Tournaments

Let’s say you’ve been playing a lot of small buy-in tournaments and your tournament skills have increased to the point where you want to start playing bigger events where you can make more money. You don’t feel ready for the major $5K and $10K events that the top pros dominate, so you want to start playing in $1K events as a stepping stone to the majors.

Let’s also assume that you’ve been beating the small buy-in tournaments at a rate of well over 200%, and you believe you would have an advantage of at least 100% in these bigger $1K events. You’ve been building your bankroll with these small buy-in tournaments, and your plan is to start using the money you’ve won to advance. The main question you have: Is your bankroll really big enough to withstand the greater fluctuations you'll encounter in these $1k events?

If you do not have a sufficient bankroll to enter $1K events at this time, that does not necessarily mean that you must resign yourself to smaller buy-in tournaments. In fact, if you are a skillful satellite player, you can start entering $1K tournaments via satellites with a smaller bankroll. As the charts above show, satellites can very effectively lower the buy-in costs of bigger events. And a lower buy-in cost means a smaller bankroll is required. But how much smaller?

Before we can figure out how much satellite skill can lower your bankroll requirements, however, let’s quickly review what your bankroll requirements would be for a given type of tournament if you pay the full buy-in cost.

Let’s say you would like to play a hundred $1K tournaments in the next year. If you live in Las Vegas, this is easily accomplished, as Bellagio has two $1K buy-in tournaments every week. $1K events are also popular preliminary events for WSOP Circuit series, WPT events, and many other special tournament series that poker rooms run throughout the year. In order to estimate the bankroll requirements for entering a hundred $1K events, assuming a player has a 100% advantage on the field, let’s use a real-world example.

At the recent WSOP Circuit Events that were hosted by Harrah’s Rincon in San Diego, the $1k event on February 13, 2007, had a total of 89 players who paid $1060 for a seat. Nine spots were paid. This was the payout structure:

PlacePayout
1st $31,079
2nd $17,266
3rd $9,496
4th $6,906
5th $6,043
6th $5,180
7th $4,317
8th $3,453
9th $2,590

Now, we have something to work with. Obviously, every $1K tournament you enter will not have this payout structure, but for our purposes, we’re going to assume that you want to know the bankroll requirements for entering a hundred of these specific tournaments. Since we already said that you estimate that you have a 100% advantage in these events, we next have to make some assumptions as to how that 100% advantage will be realized.

When we say that you have a 100% advantage, we mean that for every $1K you pay to get into these tournaments, you will cash out $2K. That cash out will pay you back your $1K buy-in and provide a $1K profit, which is a 100% advantage. From looking at the payout schedule, we can see that there is no payout of exactly $2,120 (twice the buy-in/entry) for any finishing position. Even 9th place pays $2,590, which is a 144% profit. In order to realize a 100% advantage, we will accomplish this by having many finishes with no return, but a number of finishes that return greatly in excess of 100%. (And note that many tournament pros enjoy advantages in excess of 200% and even 300%. We are using this 100% example for a player who is just moving up to $1K events from smaller events, and who is still sharpening his skills.)

Assuming you play a hundred of these tournaments, let’s create a win record that would provide an advantage in the neighborhood of 100% overall. To do this, I’ll assume that you finish in the money 16 times, or about once every 6 to 7 tournaments. Here’s a set of finishes that includes 3 first places, 3 second places, 3 third places, 2 fourths, 2 fifths, 1 sixth, 1 seventh, 1 eighth, and 84 finishes out of the money, and would earn you 100.35% on your total investment:

Finish Payout # Cashes Total
1st $31,079 3 $93,237
2nd $17,266 3 $51,798
3rd $9,496 3 $28,488
4th $6,906 2 $13,812
5th $6,043 2 $12,086
6th $5,180 1 $5,180
7th $4,317 1 $4,317
8th $3,453 1 $3,453
9th $2,590 0 $0
10-89h $0 84 $0
Total Cashed: $212,371
Total Invested: $106,000
Total Win: $106,371
Win %: 100.35%

That’s close enough to 100% for our purposes. Obviously, no player could estimate that these will be his exact finishes in 100 tournaments. We're just creating a set of finishes that would return approximately 100% profit to the player. There are many other ways that a 100% advantage could be realized. There could be fewer than 16 money finishes, but more with the higher payouts, or there could be more than 16 money finishes, but fewer with the big payouts and more low-end finishes. The above set of payouts is just one way that a player with a 100% advantage might realize this profit.

In fact, despite the assumed 100% advantage, over the course of these 100 tournaments, a player in real life would be highly unlikely to show a result this close to an actual 100% profit. His real-world result in a series of 100 consecutive tournaments would be subject to what statisticians call standard deviation. He may have an exceptionally good run of tournaments, or an exceptionally bad run, just due to normal fluctuations in the cards and the situations he encounters.

In the Appendix to The Poker Tournament Formula, pages 328-340, there is a discussion of what standard deviation is, what it means to a gambler, and how you figure it out for poker tournaments. I’m not going to reproduce that discussion here, so if you do not understand the concept of standard deviation, and especially how it applies to poker tournaments, read that chapter the book. Also, different payout schedules caused by different field sizes will have a major effect on standard deviation, so don’t assume that the discussion about this specific $1K tournament would apply to all $1K tournaments. This tournament is just an example from real life.

Using the method described in The Poker Tournament Formula, and applying it to the player with the 100% advantage in the $1060 tournaments described above, we find that although our expectation is to win (profit, after subtracting our buy-in/entry fees) a total of $106,371 over the course of 100 tournaments, one standard deviation on that result is $62,853. So, if we finish two standard deviations below our actual expectation (and 2 SDs = $125,706), we could actually finish these 100 tournaments with a loss of $19K, despite our 100% advantage!

This may sound impossible, but keep in mind that we only expect to win $106K and that $93K of our total return comes from just three first place finishes. A few bad beats and cold decks at crucial times at the final table, or before we get to the final table, can wreak havoc with our overall results.

Since we could conceivably suffer a net loss of $19K over the course of these 100 tournaments, and still be within the realm of what a statistician would consider a “normal” fluctuation, we might conclude that a bankroll of $20K would be sufficient to finance our play, though we could conceivably lose it all. In fact, a bankroll of $20K would usually be more than sufficient to finance this level of play, assuming we are correct about our 100% edge. The standard deviation on our expected results does give us a pretty good idea of the kinds of fluctuations that are possible due to bad luck in tournaments of this level over this period of time.

Technically, your $20K bankroll would probably be very safe if you cut back on the cost of the tournaments you entered if you started experiencing significant negative results. In other words, if you finish out of the money in the first ten tournaments you enter—and this is entirely possible—then you really would be wise to start entering $500 tournaments until you hit a few wins (or just one good win) to build your bankroll back up. Fluctuations of greater than two standard deviations happen all the time.

(Cutting back on the size of your bets after bankroll reductions is a method of “Kelly Betting,” another term that would be known to any serious blackjack player, but few poker players. I’ll discuss Kelly betting approaches for tournament players in more detail later in this article.)

If you are familiar with the statistical concept of standard deviation as discussed in The Poker Tournament Formula, you are probably aware of the fact that a statistician expects a fluctuation of greater than two standard deviations 5% of the time. Which is to say that if 20 players were playing these tournaments with a 100% advantage, 19 of them would expect to finish 100 tournaments within two standard deviations of their expectation, but one of them would expect to experience a fluctuation of greater than two standard deviations from his expectation.

A fluctuation of 3 standard deviations is extremely rare, as results this far from expectation have only about a 1/300 chance of occurrence, so it’s not really necessary to maintain a bankroll to withstand this much of a fluctuation. But to be safe, just based on the standard deviation, I’d advise a bankroll closer to $30K for these $1K events, assuming you have a 100% advantage.

But What About Different Levels of Aggression in Poker Tournaments?

Let’s consider the fact that we’ve created our hypothetical 100% advantage in this tournament by devising a specific set of in-the-money finishes that would result in this win rate. In the real world, there are many different ways a player could end up with a 100% advantage. He could be a very aggressive player who had fewer cashes but more top-end finishes, or he could be a more conservative player who had a greater number of cashes, but more low-end finishes. Might not the bankroll requirements for these player types differ from each other?

Let’s analyze and compare the requirements for these different player types.

The More Aggressive Hypothetical Tournament Player

To provide this player with a 100% advantage, let’s say he finishes in the money only 10 times in 100 tournaments (instead of 16 times as in our prior example), but with more high-end finishes. Here’s the aggressive player’s chart:

FinishPayout# CashesTotal
1st $31,079 4 $124,316
2nd $17,266 4 $69,064
3rd $9,496 2 $18,992
4th $6,906 0 $0
5th $6,043 0 $0
6th $5,180 0 $0
7th $4,317 0 $0
8th $3,453 0 $0
9th $2,590 0 $0
10-89h $0 90 $0
Total Cashed: $212,372
Total Invested: $106,000
Total Win: $106,372
Win %: 100.35%

Conveniently, this very different set of cashes provides this aggressive player with the same 100.35% win rate as the player in our first example.

The More Conservative Hypothetical Tournament Player

For further comparison, let’s also create a sample player who has more final table finishes (25, instead of 16 or 10), with more low-end finishes, but still, with a 100% win rate. Here’s this more conservative player’s chart:

FinishPayout# CashesTotal
1st$31,0792$62,158
2nd$17,2662$34,532
3rd$9,4963$28,488
4th$6,9063$20,718
5th$6,0433$18,129
6th$5,1803$15,540
7th$4,3174$17,268
8th$3,4533$10,359
9th$2,5902$5,180
10-89h$0 75$0
Total Cashed: $212,372
Total Invested: $106,000
Total Win: $106,372
Win %: 100.35%

In order to compare the bankroll requirements of these three different player types, who all have the same overall win rate in the same tournament, let’s use a different statistical method of analysis, the Gambler’s Ruin Formula, or, as gamblers today more often call it, Risk of Ruin (or RoR).

First, here's an explanation of what we’re trying to figure out here. A player wants to play tournaments that have a specified buy-in/entry cost, say, $1060. He knows from the above discussion on standard deviation that he could conceivably lose his bankroll due to negative fluctuations, even if he has a 100% overall advantage. The player wants to minimize his risk, so he wants to know how much of a bankroll he’d need to insure himself of a 90% chance of success, or 95% chance of success, or even 99% chance of success. Also, if the player could maintain his 100% win rate while using either a more aggressive or more conservative strategy, how would this affect his chance of success?

The Risk of Ruin Formula that was used to analyze the three sample players described above was first published by Math Boy and Dunbar in the Fall 1999 issue of Blackjack Forum. The article is titled, “Risk of Ruin for Video Poker and Other Skewed Up Games”. You may follow this link to get to the article in our online library, so we're not going to print the Risk of Ruin formula here.

Let’s just look at the RoR data on our three player types. Remember, the conservative player makes it to the final table 25 times in 100 tournaments, but wins the fewest top-end prizes. The middle-of-the-road player has 16 money finishes out of 100 tournaments, with more at the top-end. The aggressive player has only 10 money finishes, but takes four firsts, four seconds, and two third-place payouts. These are their bankroll requirements for various levels of risk:

RoR Conservative Bankroll Middle-of-Road Bankroll Aggressive Bankroll
1% 37,213 45,449 55,853
5% 24,208 29,565 36,333
10% 18,607 22,724 27,926

We want to emphasize here that the “conservative” and “aggressive” styles we've created are for purposes of analysis—they are not meant to be realistic. With regards to the aggressive player, it’s unlikely that a player who was skillful enough to always make the top three when he finished in the money (with 80% of those finishes in the top two) would never finish in any other final table position. Even the most aggressive and skillful players will suffer bad beats and cold decks and hit lower payouts occasionally.

It’s really more likely that such a player would have a range of money finishes at all levels, even if he had more than his share of the top prizes. The problem we faced in creating this player, however, was that we were attempting to maintain that 100% win rate for purposes of risk of ruin comparison, and if we start scattering a more realistic set of smaller wins among his finishes, his win rate will climb dramatically (as it tends to do in real life with the best aggressive players).

The conservative player’s results were similarly skewed. It is highly unlikely that a player with so few top-end finishes would be able to hit the money often enough to have a 100% win rate. But the purpose of the example is simply to show that playing style does have an effect on a player’s bankroll requirement, all other things being equal. If we look at the middle-of-the road player’s bankroll requirement for a 5% RoR (95% chance of success), we see he needs a bankroll in the neighborhood of $30,000. The conservative player might get away with a bankroll of about $5000 less than this, but the aggressive player might need $5000 more.

These numbers may surprise many tournament players who have not read The Poker Tournament Formula, as there is widespread ignorance among poker players with regards to bankroll requirements. On one of the WPT shows, for example, a sidebar feature showed professional poker players being asked for advice on bankroll requirements for tournament players, and wound up providing the specific recommendation that tournament players should have a bankroll of ten times their buy-in cost. Don’t we wish.... In fact, that was potentially disastrous advice for serious poker tournament players.

One other thing we must note here is that this sample tournament we’re analyzing had a prize pool based on a total of 89 players. In fact, if you are entering $1K tournaments with 300 players, or 2000+ players as in some of the 2006 WSOP $1K events, the top prizes will be much bigger, and so will your fluctuations and bankroll requirements. We don’t want you to think that a $30K bankroll is sufficient for all $1K tournaments. Again, we ’ll refer you to the detailed discussion in The Poker Tournament Formula on how field size affects bankroll requirements.

Now let’s look at how we can use satellites to lower this bankroll requirement.

Lower Buy-In Costs Mean Lower Poker Tournament Bankroll Requirements

Since we’re discussing a specific $1K WSOP Circuit Event, let’s look at one of the actual satellites that was being offered at Harrah’s Rincon that allowed a player to win entry into this event. They ran ten-player satellites that cost $240, and the satellites paid two winners. Each winner received two $500 chips and $100 cash. So, if you were one of the two satellite winners, you’d have been able to cover your $1060 buy-in/entry to the $1K tournament, and still have $40 cash to put in your pocket.

Here’s a chart that shows this satellite’s dollar value and player advantage, based on a player’s expectation of winning twice out of every 10, 9, 8, 7, 6, and 5 satellites entered:

WSOP Circuit Satellite,
$240 Buy-in, Two Winners
Each Get $1000 (chips) plus $100 cash
Dollar Value and Player Advantage If Player Wins Twice per:
 10 9 8 7 6 5
$ Value -$20.00 $4.44 $35.00 $74.29 $126.67 $200.00
Adv. (%) -8.33% 1.85% 14.58% 30.95% 52.78% 83.33%

Now, let’s say the same three hypothetical tournament players (conservative, middle-of-the-road, and aggressive), all with the same 100% advantage in the $1K events, are also skillful satellite players, and they each decide that they will always enter these $1K events through satellites. And, let’s assume their levels of satellite skill provide all of them with an expectation of winning twice for every seven (or, essentially, one for every 3.5) two-winner satellites they enter. How does this affect the actual cost to them of entering the $1K events? Here’s the math:

3.5 of these satellites will cost $240 x 3.5 = $840. For that $840 expense, the player gets $1K in chips plus $100 in cash. Since the buy-in/entry for the $1K event is $1060, the player can buy-in to the $1K tourney and pocket the extra $40 cash, leaving him with a total buy-in/entry cost of $800 even, a discount of $260 from the full tournament price.

This lowered cost per tournament drastically reduces the risk of ruin for each player. You might guess off the top of your head that paying $800 per tournament on average, as opposed to $1060, might cut your bankroll requirements proportionately. Since 800 / 1060 = 75.5%, you might be tempted to take the 1% RoR bankroll requirement of $45,449 for the middle of the road player, and multiply it by 75.5%, for a bankroll requirement of $34,300.

But, in fact, the effect of the satellite discount entry is quite a bit stronger than that. What you must also consider is that although you are entering these $1060 tournaments for only $800, the prize pool does not change. This means that the same win record will increase your percent advantage from 100% to 165%. This increased percent advantage will lower the bankroll requirement for a 1% risk of ruin for the middle of the road player from $45K to $30K.

(Unfortunately, figuring out the RoR for entering these tournaments through satellites is not quite as simple as just using $800 as the entry fee. The situation is similar to a parlay bet at a sports book. The satellite has its own flux, and that must be included with the flux on the main tournament in order to correctly calculate overall flux and risk of ruin.)

Let’s look at the RoR bankroll requirements assuming these three players always enter these $1060 tournaments through these $240 two-winner satellites, with each winning two out of every seven of the satellites:

RoRConservative BankrollMiddle-of-Road BankrollAggressive Bankroll
1%19,35830,38042,905
5%12,59219,76227,910
10%9,67915,19021,452

What Risk of Ruin Should a Poker Tournament Player Consider Safe?

Many players might say off the top of their heads that they’d be comfortable with a 90% chance of tournament success. But bear in mind that ruin means ruin. If you have a $30K bankroll, and you find yourself in that unlucky 10% of players who would expect to lose it all if playing at this risk level, then you are flat broke. So, if this $30K represents your life savings, that would be a foolhardy way to play. If it represents money that is easily replaceable (say by cashing in a few CDs when they mature), then it may not be a bad gamble.

Most professional gamblers prefer to use a “Kelly betting” system. Those of you who have read any of my blackjack books know what this is. If you are not familiar with the term, it essentially means that you always bet proportionately to your bankroll in order ensure that you never go broke.

For example, if a player was playing these $1K tournaments with a 5% RoR, he could do so with a $30K bankroll. If this same player decided in advance that he would enter tournaments with smaller buy-ins than $1K if he lost a significant portion of his bankroll (until he built it back up), then his actual RoR would be quite a bit lower than 5%. This is why, in The Poker Tournament Formula, I advise: "...Should your bankroll go into a nosedive, be willing to start entering tournaments with either smaller buy-ins or smaller fields of players, until you rebuild your bank."

In an ideal world, a player would create a chart of optimal entry fees to pay, based on his current bankroll, that would virtually eliminate any Risk of Ruin. For example, If he lost 40% of his starting bank, he would play in $600 events. If he lost 50% of his bankroll, he would play in $500 events. With a 75% loss, he would play in $250 events, etc. Unfortunately, in the real world, is that in the real world we can’t always find tournaments priced to our needs. If I lose 10% of my bankroll, can I find a $900 tournament?

For any poker tournament player on a limited bankroll, however, it makes sense to follow such a plan as closely as possible. If your bankroll drops by 10% from the amount sufficient for $1k tournaments, you have to decide if you are willing to accept the increased risk of ruin inherent in continuing to play $1k events, or if you had better drop to $500 events, if that is all that's available below the $1k buy-in level.

Also, be honest with yourself about the actual size of your tournament bankroll. Your playing bankroll should not include your rent money, car payments, living expenses, credit card or installment loan payments, or the like.

Poker Tournament Satellite Frequently Asked Questions

Q: If I’ve already played a few satellites without winning, should I still buy-in to the main event if there are no more satellites?

A: If you have the bankroll to enter the main event at full price, and you have the skill to make money in these types of events, then by all means, buy-in for the full price. If the reason you are playing satellites is to cut your costs because you can’t afford to play the bigger events, then do not buy-in for the full price. Just keep developing your satellite skills and play only in the big events when you win your seat through a satellite.

Q: If I’m just an amateur but I really want to get into a WPT main event just to play with the pros and take my shot at fame and fortune, but that $10K buy-in is a bit steep for my wallet, should I set a limit to the number of satellites I’ll play in an attempt to enter the event?

A: If you are not a skillful satellite player and you are simply attempting to enter a major event cheaply on a long shot gamble, then you should definitely decide beforehand exactly how much you’re willing to spend on satellites, and quit if you hit your limit.

Q: If I’m skillful at both satellites and regular tournaments, should I limit the number of satellites I’ll play for any one event?

A: For a skillful player, it’s a different situation. First of all, if you fully intend to enter the main event regardless of your satellite result, then there is the practical consideration of time. If an event starts at 2pm, and satellites start at 8am, it may not be in your best interest to play satellites for six hours prior to starting day one of a tournament that might go 12-14 more hours.

Some pros always play a satellite or two before major events, even when they can afford the full buy-in price, because the satellites are a good value and can be used to lower their overall tournament expenses. If a pro can win one out of seven $1K satellites in order to enter $10K tournaments, and if he plays just one satellite before each major event, then six times he’ll be paying $11K for his seat, and once he’ll pay just $1K. If he plays 21 of these $10K tournaments per year, his three satellite wins will lower his overall tournament cost (and raise his overall profit) by $9K.

If time constraints and guarding against fatigue aren’t part of the equation, and you have an edge at satellite play, there is no reason to limit the number of satellites you’ll play to enter any one event. During big multi-event tournaments like the WSOP, there are satellite pros who virtually camp in the satellite area, playing one satellite after another, day after day.

On some days, they may play ten satellites without a single win, then they’ll win three or four the next day. They use the tournament chips they win to buy-in to the events they want to play, and sell the rest to players for full value. Obviously, these pros pay no attention whatsoever to limiting the number of satellites they’ll play for any one event. Nor should they. They’re in it for the long run, and if they have the skill to beat the satellite fields and the house edge, they’ll come out way ahead in the long run.

Conclusion

If you are serious about playing in major tournaments with big buy-ins, there is a huge value to developing satellite skills. At the 2006 WSOP, I overheard one player commenting to another that he was surprised at how many of the big name pros were entering satellites, since they could so easily afford the full buy-ins. If you look at the value of satellites as shown in the charts provided in this article, you can see why many pros are attracted to satellites.

If you play a lot of major events, and a lot of satellites to enter these events, you can substantially lower your overall annual tournament costs while increasing your percentage return. I don’t care how much money you have. If you’re paying $10,000 for an event that you could get into for $8,000 (or less), you may be a great poker player, but you’re not that great at financial planning.

Notes and Acknowledgments

Much of the material in this article will be unfamiliar to poker players who have not read The Poker Tournament Formula, because bankroll requirements are estimated using statistical methods that are not taught in your everyday high school math courses. Professional gamblers really need to understand this math, or they will condemn themselves to many years of going broke repeatedly, no matter how skillful they are. As I put it in The Poker Tournament Formula (Chap. 28, “How Much Money Do You Need?”):

It’s the rare blackjack book these days that doesn’t provide at least some information on such topics as standard deviation, the Gambler’s Ruin formula, risk-averse betting strategies, the Kelly criterion, and various related topics, in addition to simplified charts of data that card counters can use to estimate their bankroll requirements.

Most poker books, by contrast, stick to strategic advice exclusively. Blackjack players learn early how to manage their bankrolls; poker players learn early how to hit up their friends when they go broke.

Thankfully, some five months after I published those lines in The Poker Tournament Formula, another poker book has been published that does deal seriously and intelligently, and in much greater depth for cash game players, with the topic of bankroll requirements. This book is The Mathematics of Poker , by Bill Chen and Jerrod Ankenman. This book is to poker what Peter Griffin’s The Theory of Blackjack is to that game.

Unfortunately, like Griffin’s book, much of the material in the Chen/Ankenman book is not readily accessible to a player who has not taken some college level courses in probability and statistics. I still urge any serious poker player to get this book, just as I have always recommended Griffin’s book to all serious blackjack players. The Chen book is essentially a book for math heads, but there's a lot of discussion on a wide range of topics I haven't seen elsewhere for serious players. He even delves into risk of ruin when you don't know your advantage.

Risk of ruin is a statistical measure that blackjack players would be familiar with, but that has been long absent from the poker literature. There is an excellent explanation of RoR in Dr. Allan Wilson’s classic text, The Casino Gambler’s Guide (1965). But neither Wilson’s description of risk of ruin, nor any of the descriptions that have been published in many blackjack texts since then, allow for the formula’s use in a game where there are multiple possible payouts, ranging from a loss of the bet (buy-in), to a modest win (low-end finish) to a very large payout compared to the size of the initial bet.

The first published discussion of RoR that I know of for games with multiple payout possibilities was an article by Russian mathematician Evgeny Sorokin that appeared in the March 1999 issue of Dan Paymar’s Video Poker Times newsletter. In response to Sorokin's article, professional gamblers Math Boy and Dunbar developed an Excel spreadsheet method of applying Sorokin’s generalized risk equation to virtually any game with a skewed payout structure, and I published their method in the Fall 1999 issue of Blackjack Forum, in their article, “Risk of Ruin for Video Poker and Other Skewed Up Games”. Now, Math Boy has helped me to adjust his method for analysis of poker tournaments (or any other gambling tournaments).

I am especially indebted to Math Boy for creating an Excel spreadsheet for me that would not only estimate a tournament’s standard deviation and risk of ruin, but would automatically recalculate these values based on entering the tournament via satellite, at any satellite cost, and with any selected percentage of satellite wins. The spreadsheet is not currently user-friendly for anyone who is not familiar with some of Excel’s advanced statistical functions, but Math Boy is working on a simpler version, similar to his Patience Factor Calculator, that he plans to make available to players at this Web site in the near future.

Incidentally, a number of players have asked me where I came up with the standard deviation formula that appears in The Poker Tournament Formula as they had never seen a method for calculating standard deviation for a game with a tournament payout structure. My method was simply to modify a formula originally created by Doug Reul, which first appeared as the “Volatility Index” in one of Dan Paymar’s 1996 issues of Video Poker Times, and can currently be found in his book, Video Poker: Precision Play  ♠

For more information on poker tournament theory and strategy, see the Poker Tournament Formula Home.

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