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Arnold Snyder shows in this article the flaws in the mathematical model used by Mason Malmuth to purportedly refute the argument behind the rebuy advice in The Poker Tournament Formula. Read this article to understand why David Sklansky's advice on rebuy strategy is wrong, and to understand why the poker tournament rebuy strategy in Snyder's book is right.
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Poker Tournament Rebuy Advice

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Response to Mason Malmuth on the Rebuy Advice in The Poker Tournament Formula
By Arnold Snyder
(From Blackjack Forum , Summer 2006)
© Blackjack Forum 2006

[Editor's note: This article is a response to a post by Mason Malmuth at his Web site in which he purports to refute the argument behind my rebuy advice in The Poker Tournament Formula. —A.S.]

A player recently alerted me to an argument that Mason Malmuth had posted at his twoplustwo web site which purports to refute the logic I present on optimal rebuy strategy in Chapter Ten of The Poker Tournament Formula.

Mason quotes sections from my book, then presents his contrary opinion. First, Mason describes how I show that in a coin-flipping contest, where neither player has an advantage, if one player starts with twice the chips of the other player by paying twice as much for those chips, then neither player has an advantage. Mason agrees with me on this point.

He disagrees, however, with my analysis of the effect of a player advantage when one player makes a rebuy. Here’s what Mason says:

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….

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.

Well, this certainly sounds like a great argument, Mason. You allege that this 60-40 payout structure is a more realistic example of a real world tournament than the winner-take-all format I used. But let’s look more closely at this hypothetical tournament that you set up for your model.

There are only two players who buy-in for $100 each. Each player gets a $100 chip for his buy-in. The total prize pool, assuming neither player makes a rebuy, is $200, and the 60-40 payout structure ensures that the winner will get $120 and the loser will get $80.

So, in your 60-40 tournament, the most either of these players can win is $20, and the most either player can lose is $20. In fact, there really isn’t any reason for these players to be buying in for $100, since what happens at the end is that each player is immediately refunded $80 of his $100 buy-in, while the tournament winner gets the $40 remaining in the prize pool, the only money that is being contested.

Your 60-40 tournament is really a $20 buy-in tournament for which each player gets a $100 chip to play with. You do note that Player A, who has the 10% advantage in this hypothetical 60-40 split tournament, assuming no rebuy, has an actual dollar return expectation of exactly $2 per tournament, so I would think that this might have given you a clue that if he has a 10% advantage and a $2 expectation, he’s actually in a $20 buy-in tournament, but somehow this escaped you.

Then, you analyze what would happen if Player A made a rebuy for $100 to get a second $100 chip! In other words, his first $100 chip costs him only $20, but his rebuy chip costs him $100! In your tournament, he must pay five times more for his rebuy chip than he pays for his initial buy-in!

And, you come to the conclusion that if he makes the rebuy in this tournament, instead of a $2 win, he will have a $36.50 loss. But what does this have to do with any real world tournament? I’ve never played in or seen any tournament where all players were guaranteed to get 80% of their initial buy-in back even if they lose, and I’ve never heard of a tournament where the cost of the rebuy chips is 500% of the cost of the initial buy-in chips. I especially like the part of your argument where you say:

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 then you provide a model that has absolutely nothing to do with the real life situation. In your attempt to refute my rebuy advice, you state that the player who rebuys will lose $36.50 simply because it’s not a winner take all format. No, Mason, he loses that much money because he’s paying 500% of the cost of his initial buy-in for his rebuy chip. His cost per chip is substantially larger than his competitor’s cost per chip, and he’s increasing his competitor’s prize far beyond what his measly 10% advantage will deliver.

In fact, it’s not difficult to create unrealistic models in which my rebuy strategy would be wrong. This would be true, for instance, if Player A were the only player making the rebuy in a small-enough multiple-player tournament, especially if Player A’s advantage over his competitors was only 10%, as in my PTF example.

In a four-player coin-flip elimination tournament, where each player buys in for $100, and only Player A makes a $100 rebuy for an extra chip, while Players B, C, and D make no rebuy, with the $500 prize pool going only to the top two players, divvied up 60-40 ($300 and $200) between first and second place, it can be shown that Player A would lose on average $2.45 per tournament even with his 10% advantage.

But an example like this has nothing to do with real world tournaments. In single-rebuy multi-table tournaments, the vast majority of players make the allowed rebuy. For example, in the Orleans Friday and Saturday night single-rebuy events, 95+% of all players make the allowed rebuy. From my experience, that level of rebuy participation is the norm. In multiple rebuy tournaments, most players make multiple rebuys. At Orleans, the average player in their multiple-rebuy tournament makes three to four rebuys. In the WSOP $1K rebuy event, in which rebuys are not discounted, the average player makes two rebuys. Also, in real-world multi-table tournaments, skilled players are not playing with puny 10% advantages. They play with advantages ranging from 100% to 300% or more. I used 10% in my coin-flip example just to keep it simple and explain the logic.

In my description of “rebuy maniacs” in PTF, I do explain that it is possible to make too many rebuys by playing too loosely, because you would be buying too many chips for your competitors and you would be unlikely to have a sufficient skill advantage to overcome the cost you are paying for your own chips.

The coin-flip examples in Chapter Ten are provided to make simple points of rebuy logic that would be true in almost all real-world tournaments. Further, I tried to isolate these points in such a way that non-mathematicians could grasp the logic. I have an example that shows that discounted chips can create an advantage for a player by lowering his cost per chip. I have an example that shows that if a player has an advantage over his competitors then even full-price rebuy chips can raise his dollar return. And I point out that less-skilled competitors cannot nullify a skilled player’s advantage by purchasing rebuy chips themselves. I even have an example that shows that if the rebuy chip purchase funds do not go into the prize pool—as in the case of “dealer bonus chips”—the purchase of the chips will still in most cases add value to the skilled player.

So, you have provided not only an unrealistic mathematical model, but also a tournament format that is completely irrational. To be sure, there are a number of real differences between a coin-flip model and a real-world tournament. But these differences tend to enhance the value of rebuys for skilled players. For one thing, in real-world multi-table tournaments, the vast majority of the players (generally 90%) will not finish in the money at all.

Also, I pointed out that one of the real-world tournament factors that a coin-flipping contest does not mimic is the intimidation value of chips. In real tournaments, players with more chips more easily steal pots without confrontation. Any player with any significant amount of real-life experience in tournaments will immediately recognize that this is true. This means that having more chips actually raises a player’s percentage advantage, even when that player has the same level of skill. Your model fails to address this factor. My model also leaves out the intimidation value of chips, but I take care to mention this real but difficult-to-quantify factor as it raises the mathematical value of more chips beyond what we can deduce from coin-flip logic.

Another difference between the coin-flip examples and a real-life tournament is that the coin-flip examples consist of only a few bets, with one player all-in on every single flip. In a real tournament, your extra rebuy chips will force unskilled competitors to play against you for many hours, not for just a few flips. The longer a skilled player can keep unskilled competitors playing against him, the greater the dollar value of his chips.

In conclusion, I want skillful players to be aware that they should follow the rebuy advice in The Poker Tournament Formula. The logic of my examples is valid, and David Sklansky is incorrect in his Tournament Poker for Advanced Players when he writes: "I think a decent rule of thumb would be to add on if you have less than the average number of chips at that point, and not otherwise." Sklansky and I both state that discounted rebuy chips should almost always be purchased by skilled players, as they will lower your cost-per-chip and raise your overall dollar return. But what Sklansky's advice shows that he does not understand, but what I show in The Poker Tournament Formula, is that having more chips also forces your less-skilled opponents to give you more action, and that this gives you a greater dollar return on your skill.

Sklansky's advice also shows that he doesn't understand the strategic and psychological advantage of a bigger stack, and its effect on increasing your percentage advantage. Always buy as many chips as you can, as soon as you can. And remember that chips have an intimidation value that raises their mathematical value beyond their cost. When you have the chips, use them. They allow you to afford greater variance and you should take advantage of this. In no-limit tournaments, chips are a major weapon. There are a few exceptions to these general rules, when rebuy chips ought not be purchased, and these exceptions are discussed in the book.

If a tournament ever materializes where all players are refunded a large portion of their initial buy-in, or where rebuy chips are sold for 500% (or any other percent greater than 100%) of the cost of the initial buy-in chips, I’ll be happy to provide a detailed analysis.  ♠

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