Multi Qualifier Satellite Tournaments

With entry fees in major events ranging from $5,000 to $25,000 these days, a mini-industry of satellite tournaments has arisen, to give players a shot at the big money for a modest investment. In return for a relatively small entry fee, you get a chance for a free entry to a major event, plus some money toward expenses. In many cases the satellite tournaments are open-ended. All comers can play, and the organizers will give away as many free entries as they can, based on the number of participants. Satellite tournaments are a great deal for players, and in fact many top players enter satellites routinely as a way of reducing their overall expenses on the circuit. (When I won the World Series in 1995,1 had qualified from a live super-satellite with a $100 entry fee. In the past two years, both Chris Moneymaker and Greg Raymer qualified from online satellites.)

The early part of a satellite tournament works just like any other tournament. You play your cards, make moves, and accumulate chips. The endgame, however, is quite different. The top few finishers get an identical big prize, while no one else gets anything. Being on the bubble is no longer a difference between zero and a modest prize; it's the difference between everything and nothing.

Endgame decisions in these multi-qualifier events can become quite unusual and counter-intuitive. Consider the following simple situation.

Example No. 1. Six players remain in a satellite qualifying tournament. The final four places advance to the big tournament. Fifth and sixth places receive nothing. Of the last

six players, Players A, B, C, and D all have 4,500 chips Players E and F have 1,000 chips each.

Question: At this point, what is the probability that each plaver will qualify?

Answer: This is pretty easy to calculate if we make a couple of observations.

First, notice that when we get down to four players, each of the remaining players will have a 100 percent chance of qualifying. Therefore at any earlier point, the sum of all the probabilities of qualifying must add to 400 percent (not 100 percent).

Second, whatever the chance of qualifying may be, Players A, B, C, and D must all have the same chance, and Players E and F must also have the same chance. (To make the problem easier, we'll ignore for now that whoever is on the button next hand will have a slightly better chance than the others, and so forth.)

Third, the chance of qualifying at any particular time is approximately proportional to the chip counts of the players.

Now let's put all these observations together with a little simple algebra. Let's say that E's chance of qualifying is "x." Then F also has "x" chance of qualifying, while A, B. C, and D all have 4.5 times "x." So we can write down a little equation:

So 20x = 400, and x = 20. Therefore players E and F each have a 20 percent chance of qualifying, and Players A. B, C, and D all have 4.5 times as much chance, or 90 percent each. And the probabilities all add to 400 pcrccnt, as they should if four people are going to qualify.

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These numbers all make sense. E and F are clearly big underdogs to make it, but they're not out of the running. A, B, C, and D are huge favorites, but not certain to get in.

Having established each player's chance of qualifying a priori, let's now sit them down at the table and watch what happens.

For the next hand, the players are arranged as follows:

Sm Blind Player C $4,500

Big Blind Player D $4,500

1 Player E $1,000

2 Player F $1,000

The blinds arc $100/$200. Players E and F fold. Player A goes all-in. You are Player B. You pick up

Is your hand good enough to call?

Discussion: Your first reaction should be annoyance, since Player A clearly doesn't understand that the idea for the leaders is to gang up on Players E and F, not to bash each other. Just his bad luck that he happened to pick the wrong hand to go all-in on ... or did he?

Before you reflexivcly shove all your chips in the pot, contemplate your winning chances against some hands that

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