Poker Tournament Formula Revisited

[Mason Malmuth]

Hi Everyone:

The following comes from one of the posts in The Poker Tournament Formula thread by Arnold Snyder's associate Radar.

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They have to show that Arnold's math is wrong in Chapter 10 of The Poker Tournament Formula, where he shows the mathematical basis of the edge a big chip stack has over a small chip stack in a tournament.


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Well, let's take a look at it. In chapter 10 of The Poker Tournament Formula Arnold Snyder addresses the idea of rebuys and add-ons. Here's a little of what he does.

First off, he examines a coin flipping contest between Player A and Player B where each player bets one $100 chip on the outcome of a coin flip. First he shows that if Player A has two $100 chips compared to Player B's one $100 chip Player A can expect to win the contest twice as often as Player B since

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Player A can win the tournament on a single correct call, whereas Player B is required to make two correct calls.

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But he also shows that Player A will not win more money than Player B since

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The two-thirds of the time that Player A wins the tournament, he will win Player B's $100 buy-in. But the 1/3 of the time that Player B wins the tournament (with two correct calls) he will win both Player A's buy-in and rebuy for a total of $200.

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In other words neither player has an advantage, and I agree.

The next step is to look at what happens if Player A has a 10 percent playing advantage. Without going through the details, Snyder now shows that Player A expects to win $10 per tournament if both he and Player B each have one $100 chip. Again I agree.

Then it gets a little more interesting. Snyder now has Player A start with two $100 chips and Player B sticks with his one $100 chip. Since A has a 10 percent playing advantage, we expect him to show a profit, but what happens is that his profit now increases to $17.50 per tournament (as opposed to the original $10) since the average tournament will now last longer because Player B must win twice in a row to win the tournament. Thus it's pretty clear that the more chips Player A has the larger his expectation will be since he is the better player.

This is an interesting result and I don't remember seeing it written up in any other book in the poker/gambling field. It also leads Synder to all sorts of conclusions. We are told to:

1. Always rebuy and add on regardless of how many chips you may have.
2. To rebuy as soon as possible (even if you are just one chip below the rebuy threshold).
3. To beware of chip dumping, especially at the final table.
4. To always play very aggressively.
5. And many other plays which I disagree with when you have a lot of chips.

So what's happening here.

Well, in the world of mathematical statistics, something that I use to do professionally many years ago, it's important to have the problem well defined. Put another way, when doing mathematical modeling, you would like a model (such as a coin flipping contest) that is simple to understand but at the same time does a pretty good job of representing the more complex phenomenon (such as a poker tournament). If this is the case, you can often draw valid conclusions about how to proceed in the more complex situation. And, this is where I have a problem with The Poker Tournament Formula.

The model that Snyder is using does a pretty good job of representing a winner take all poker tournament. It does not do a good job of representing a percentage payback poker tournament where the prize pool gets divided up among many players, and most of today's poker tournaments are of the percentage payback structure.

Let's go back to Snyder's coin flipping model where Player A has a 10 percent playing advantage over Player B, but this time the winner of the tournament gets 60 percent of the prize pool and the loser gets the remaining 40 percent. (I think everyone will agree that this more accurately represents what happens in a poker tournament than the winner take all model.)

Now without showing the math, the expectation for Player A is $2 when both he and Player B each start with one $100 chip. Notice that this is not as good as the original $10 expectation as before, but it is still a good bet and Player A would probably like to play a bunch of these tournaments.

Now let's suppose that Player A starts with $200 in chips meaning that the total prize pool is now $300. For him to have an expectation of $17.50 before, it means that he is winning this tournament 72.5 percent of the time. But what happens now when there is a 60-40 split?

First off, Player A will still win the coin flipping tournament 72.5 percent of the time. That's because his 10 percent playing advantage has not changed. But his expectation is now negative $36.50. Furthermore, since his original expectation was to win $2 (with only one $100 chip) the purchase of the second $100 chip (for $100) has cost Player A $38.50. This makes a huge difference since we can now see that a more accurate model does not behave in the way Snyder's original model behaved. In fact, it behaves just the opposite and clearly implies that many of the conclusions should be different.

Best wishes,
Mason