Poker Tournament Formula Revisited
 

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.

[/ QUOTE ]

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

Re: Poker Tournament Formula Revisited
 

Hey, Mason. Have you given the book a rating yet?

Re: Poker Tournament Formula Revisited
 

Mason,

Thanks for sharing this. This type of ad hoc education is another reason why so many of us are indebted to 2+2.

Re: Poker Tournament Formula Revisited
 

Yes, I think that we can safely assume that in any tournament where first prize pays less than 2 buy-ins, that we shouldn't rebuy. Please explain why this isn't a trivial case. I should think that in your example first place would have to pay more than 67.7% of the prize pool for your example to have any relevance.

I agree that Snyder's example doesn't take %age payouts into account, but your example, because you have defined Player A's EV as negative by saying that he must put in an amount that is greater than the maximum he can win, is trivial.

Re: Poker Tournament Formula Revisited
 

While your example makes alot of sense, heads-up S&G's always have one winner, and no rebuy. So if I can get a 2:1 chip advantage on my opponent with a 10 percent edge, I should show a nice profit.

If I was in a rebuy tournament where no one would rebuy except for me, I wouldn't rebuy either. I'd be fattening the prize pool while reducing my ROI. I might still add-on even if no one added on though, but only if the add-on gave me a significant chip advantage. I think the result would still be bigger prize pool, smaller ROI, unless the add-on somehow occurred when we were in the money or very close to it.

Re: Poker Tournament Formula Revisited
 

Hi Ortho:

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I agree that Snyder's example doesn't take %age payouts into account, but your example, because you have defined Player A's EV as negative by saying that he must put in an amount that is greater than the maximum he can win, is trivial.

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No. It's not trivial. The reason I used the percentages I did is that in many of today's tournaments the difference between first and second is not that great. But if you like, switch the payback percentages to 80 percent for first place and 20 percent for second and still give Player A a 10 percent playing advantage. The overall expectation for Player A is to now lose $9.50 on his $200 investment which means the expectation on his rebuy is now a negative $11.50.

For the rebuy in this example (with Player A having a 10 percent playing advantage) to be profitable for Player A first place would need to pay approximately 89 percent (or better) of the total prize pool. Also notice that if the rebuy (or add-on) is not profitable for Player A it becomes profitable for Player B.

Best wishes,
Mason

Re: Poker Tournament Formula Revisited
 

So, how many times are we going to revist and rehash the SOS with this book? Let it go already. We get it, buy 2+2 books only...right?

Re: Poker Tournament Formula Revisited
 

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We get it, buy 2+2 books only...right?

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It's our policy to recommend those books and products which are good and from which all our readers will benefit from. It's also our policy to steer our readers away from that material which is questionable. You can do a search and find many non-Two Plus Two books that have been recommended by me.

MM

Re: Poker Tournament Formula Revisited
 

In a rebuy tournament, if you're the only one who rebuys, you are the sucker.

Let's create a different scenario: a six-man tournament that pays 2/3 to 1st, 1/3 to second. In this tournament, each player buys in for $50. A dealer button moves around the table, just like in Hold'em, and the two players to the left of the button must post a $50 blind and flip a coin to determine a winner. Play continues this way until we are heads up, at which time the bet per flip increases to $100. Let's say Player A had a good run and is sitting with $200 in front of him. Player B has the remaining $100. You said that Player A should win twice as often as Player B. With that in mind, Player A wins $200 twice and $100 once for an average profit of $116.67([500/3]-50). Player B wins $100 twice and $200 once for an average profit of $83.33([400/3]-50). Player A's "chip" advantage has increased his potential profit, even without some sort of edge.

Maybe Snyder's book made some careless generalizations with regards to rebuy strategy, but his example about "chip" advantage seems to be somewhat accurate.

Re: Poker Tournament Formula Revisited
 

Why use these pointless examples of tournaments no one plays in? And, why do you think they prove any points at all?

Re: Poker Tournament Formula Revisited
 

Think about it... let's say Player A has 4000 in chips and Player B has 2000 in chips. If neither player has an edge(they both play the exact same, no adjustments), over the long run, Player A would win twice as often as Player B.

I don't see anything pointless about it... the coin flip examples are used to simulate a game in which no one would have an edge. They show how someone with more chips will be a favorite against someone with fewer chips, unless one of the players has a large enough edge and small enough blinds to overcome their chip disadvantage.

Re: Poker Tournament Formula Revisited
 

[ QUOTE ]

It's our policy to recommend those books and products which are good and from which all our readers will benefit from. It's also our policy to steer our readers away from that material which is questionable. You can do a search and find many non-Two Plus Two books that have been recommended by me.

MM

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I really don't need to do any searching. I am aware of your support for other books written with similar subject matter. It's simply a matter of how many times are we going to have to "revisit" information regarding this book.


There are a ton of other books out there that you can steer players away from, but when you start an entirely new thread, it just seems more personal than informative.

Obviously, I am a bystander to these threads as there is more inforamtion to absord daily, than I can find time to reply. It is just my opinion, which we are all entitled to, that I had to throw out a line for the dead horse being beaten. It's nice to see new ideas, whether you support them or not, in the arena of NLHE.

Re: Poker Tournament Formula Revisited
 

Hi Bang:

For your example to be accurate, you need to figure in the probabilities that Players A and B achieve the results they have, and you also need the probabilities for all other possible outcomes. Once this is done, you can then compute their overall expectation which will be to break even in your example.

best wishes,
Mason

Re: Poker Tournament Formula Revisited
 

Hi Deakon:

I see your point. But the reason I decided to revisit was the idea that this book had unique mathematics in it which could not be disputed. So I thought it was important to show that was not the case.

In addition, understanding how to properly frame problems from a probability/gambling standpoint is key to being successful at what it is all of us are interested in. So while I do sort of agree with you that The Poker Tournament Formula has been discussed enough, it still seemed important from my point of view to address this issue.

best wishes,
Mason

Re: Poker Tournament Formula Revisited
 

okay, lets say that player a has more chips than player b, is a better player and bathes on a regular basis whereas player b is a shortstacked donkey with BO. Player B might still win because Donkaments are all luck in the short term.

Only if Player A has the emotional stability to endure mindless beat after mindless beat without losing his mind over a significant number of Donkaments does his edge begin to show.

I like the coin tossing scenarios beause they reflect the huge amount of luck involved in Donkaments.

Before the flame war starts, yes, good players have an edge in Donkaments yes, Donkaments are fun/exciting to play but, if you can't handle massive suckouts, stay away from Donkament play.

Re: Poker Tournament Formula Revisited
 

Mr. Malmuth,

And, I see your point as well, sir. You and the people of 2+2 provide valuable information time and again and I value your opinion in regards to these matters of this insane game.

Thank your comments and I hope I did not come off as being offensive in any way.

Keep up the work you do and I look forward to more publications from 2+2 in the future.

Re: Poker Tournament Formula Revisited
 

[ QUOTE ]
Hi Bang:

For your example to be accurate, you need to figure in the probabilities that Players A and B achieve the results they have, and you also need the probabilities for all other possible outcomes. Once this is done, you can then compute their overall expectation which will be to break even in your example.

best wishes,
Mason

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Mason,

I want to make sure I understand your point here. It appears that you're assuming in Bang's example that the player with 4,000 chips has them due to a re-buy (although I didn't read that into Bang's example, especially in light of his comments just before that in the thred). Assuming A has the chips due to rebuying or adding on then since neither player has an edge Player A would win 2/3 of the time, but would have also contributed 2/3 of the prize pool so he would break even. I think that's what you're saying.

However if this wasn't a rebuy tournament and Player A had acquired the chips part way through the tournament (maybe by knocking out player C who isn't as skilled as either A or B) then at this point (staying with the assumption that A and B are equally skilled) then we would be correct in assessing player A's chances of going on to win the tournament to be about 2/3 or 67%, correct?

Al

Re: Poker Tournament Formula Revisited
 

Hi BigAlK:

Yes. If Player A at some point in a tournament has more chips than Player B then he is a favorite at that instant over Player B. And in your example, Player A does have a 2/3's chance to win the tournament.

best wishes,
Mason

Re: Poker Tournament Formula Revisited
 

(Haven't read Snyder's book)


I'm not sure that analyzing so in-depth in a heads-up scenario should lead us to so many conclusions regarding the value of re-buys in a multi-player field.
However, I'm not intelligent enough to know why so I readily pass the baton to those who are smarter than I am.

But I was kind of surprised to see Mason use the 60/40 example here as being more closely related to pay-out structures in an MTT.

While it's true that the difference between 1st vs 2nd is often not too huge (you'll be pretty happy if you get either) I think about the rest of the tournament pay-out structure including the bottom pay-outs where you pretty much get back your entry-fee if you limp your way into the money.

I thought that most agreed that it is best to play with a 'win at all costs' type mentality because it's higher EV than just playing a weaker game designed to barely get you into the top 10% to make it into the money.


If we assume that it's best to 'play for the win' then wouldn't we also assume something closer to a 100/0 type of structure for these little heads-up coin-flip type demonstrations?


In other words, to simulate an MTT-type pay-out structure in this coin-flipping exercise I don't know why we would look at JUST the difference between 1st vs 2nd.

I would think we should look at the difference between a 1st OR 2nd place finish compared with a 99th or 100th place finish.
And with that perspective, wouldn't a 100/0 pay-out structure apply to the heads-up coin-flipping exercise?


Perhaps there's something from the exercise that I'm missing.

Re: Poker Tournament Formula Revisited
 

Also - I'm glad that Mason is continuing to address this as I find it quite interesting and obviously a lot of 2+2'ers did the first time around as well.


He is not beating down Snyder imo. He is disagreeing with him which he is certainly allowed (and welcome) to do.


It would not surprise me if Mason said that despite some of the flaws he finds in this book that he still believes it can help many players' in their tourney play.

Re: Poker Tournament Formula Revisited
 

Hi Bob:

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It would not surprise me if Mason said that despite some of the flaws he finds in this book that he still believes it can help many players' in their tourney play.

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That's exactly what I've said. But the problem is that it frequently has you making the right plays for the wrong reason (in my opinion) and occasionally has you making negative expectation plays such as purchasing an add-on when you have a huge amount of chips relative to the field.

best wishes,
Mason

Re: Poker Tournament Formula Revisited
 

Hi Bob:

The reason I used the example I did was to mimic The Poker Tournament Formula as closely as possible. It just turns out that this is something which I addressed originally in a 1986 issue of Poker Player and you would probably feel more comfortable with the examples used. If you want to see exactly what was written, most of the original work appears in my book Gambling Theory and Other Topics.

Best wishes,
Mason

Re: Poker Tournament Formula Revisited
 

These examples are all interesting, but there are still a couple of things that make me wonder. First of all, where is the break-even point for player A (given he has a 10% advantage)? Second, what sort of advantage does player A need to make Snyder's example work? Last but not least, why doesn't anyone tell Negreanu to stop making these excessive rebuys? At least he should be forced to donate some of his winnings to the American Statistical Association.

Re: Poker Tournament Formula Revisited
 

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...occasionally has you making negative expectation plays such as purchasing an add-on when you have a huge amount of chips relative to the field.

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It's possible that some following this thread might miss this subtle distinction. In the original post Mason mentions 5 conclusions that Snyder comes to based on his coin flipping example. Although it appears that there are other parts of these conclusions he disagrees with to some degree the only item he's addressing here is #1 on the list ("Always rebuy and add on regardless of how many chips you may have") and the disagreement is with the advice to always do this.

I should preface this with the disclaimer that the discussion below is based on my interpretation of what Mason is saying in this thread. If I'm missing the point I'm sure he'll correct me. [img]/images/graemlins/smile.gif[/img]

David Sklansky has a short section in TPFAP on re-buy tournaments. In that he concludes taking the add-on is correct up to a point. The rule of thumb he gives there is to definitely take the add-on if your stack at that point is less than average. He also discusses some situations related to the structure of the add-on that might make taking the add-on correct in his opinion even if your stack is larger than average. (These special situations are when the number of chips in the add-on are significantly larger than the number of chips in the original buy-in or when the add-on is selling chips at a discount).

Mason's example taking into account % payout tournaments demonstrates why there is a point at which the extra equity gained by adding on does not increase your expected pay off enough to off set the cost of the add on. Although this does not refute Snyder's proof that more chips gives you a greater chance of winning the tournament (and isn't intended to) it does show that there is a point at which taking the add on doesn't make sense. Where this point is up for grabs although Mason has clearly shown in his simplified example that this point exists.

If we assume that Sklansky's "rule of thumb" is correct and the add-on is EV+ if you have an average stack then the point where taking the add-on ceases to be correct is no lower than that. There is a stack size large enough that the add-on is EV- although this might be so large in a typical tournament that the chances of anyone building a stack this large in the rebuy period is extremely unlikely. I assume the variables to compute where this point is would be your stack size, the number of chips in play, the field size, and the structure of the payouts. From a practical standpoint computing this at the time you're making the decision probably wouldn't be possible. However doing so away from the table for different situations might be practical to give you a feel for how the various variable interact.

My gut instinct (possibly not based in reality) is the point where taking the add on is not correct is most likely high enough that in almost every situation you're likely to actually encounter taking the add on is the right move. I'll let this idea toss around in my subconcious for a while and see if I can think of a way to substantiate my feeling.

Al

Re: Poker Tournament Formula Revisited
 

Hi Shandrax:

The break even point in this example was first place paying 89 percent of the prize pool.

As for Negreanu, it's correct to rebuy in percentage payback tournaments when you are broke, but wrong when you have a lot of chips. See my book Gambling Theory and Other Topics for more information.

Best wishes,
Mason

Re: Poker Tournament Formula Revisited
 

Hi Al:

I think you have this right. If the tournament was winner take all and you were a superior player then you should always take the add-on and rebuy as soon as you can as Snyder recommends. If the tournament is a typical percentage payback then you should take the add-on unless you are doing very well in the tournament and wait until you go broke to rebuy.

This has significant strategy implications. For instance, automatically playing the first hand in an attempt to lose one chip so that you can rebuy seems like silly advice to me in a percentage payback tournament.

Best wishes,
mason

Re: Poker Tournament Formula Revisited
 

I decided to take a closer look at this. Tournament players know that in percentage payback tournaments, every chip is worth less than the chip before it. The question is at what point do you have so many chips that taking an add-on is not worth it?

I started by looking at the coin flipping problem. Imagine a 10-player tournament that pays 50/30/20 to the top 3 spots and where everyone starts with 1 chip. 2 players are selected at random and they flip for 1 chip. I did some simulations and it turns out that a 10% skill advantage is way too high an assumption for this problem. If our skilled player wins coin flips 55% of the time, he wins the tournament 20% of the time. This is much higher than good SNG players can do; a more reasonable number is 14% wins.

So let’s take a skilled player that has a finish distribution in the top 3 spots of 14.0/13.4/12.7%. For a $100 tournament, this equates to an EV of +$35.60 (minus the entry fee). What if he bought twice as many chips and no one else bought any? His finish distribution will be about 24.6/20.3/16.4% for an EV of +$38.37 (minus entry fee) for a gain of $2.77. [img]/images/graemlins/blush.gif[/img] That percentage payback effect is pretty big. Even when everyone has average chips, superior skill barely covers the strong diminishing returns you experience with tournament chips.

What if at the end of the rebuy period he was able to build a stack equal to twice the amount of the other players? Should he now take the additional add-on? By not taking the add on, his finish distribution is still 24.6/20.3/16.4% and his EV is +$138.37. If he takes the add-on, his distribution only improves to 32.8/23.9/16.8% and his EV drops to +$96.23. That add-on costs $42.14 in profit!

Tysen

Re: Poker Tournament Formula Revisited
 

'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.)'

I'm not sure I do agree when we are talkig about rebuy MTT's.

The prize distributon is so heavily weighted in favor of the final table that the 'winner takes all' model (if by 'winning' you mean making the top 1% of the field) may well be better than assuming a HU SNG with a 60/40 split.

Of course neither is perfect - I wouldn't know where to start running EV simulations for a 1000 person MTT with rebuys, but I suspect when working out whether or not to rebuy the winner takes all approach is going to get closer to how a large MTT operates in the first hour with a big field...

Also I think that whoever is taking the maintinable 'edge' from the results of folks who are playing say 10 SnG's at once may be undestimating how much dead money there is in big online MTT's. An edge of 5% doesn't seem that outrageous to me - but that's an argument for another day..

Re: Poker Tournament Formula Revisited
 

[ QUOTE ]
Hi BigAlK:

Yes. If Player A at some point in a tournament has more chips than Player B then he is a favorite at that instant over Player B. And in your example, Player A does have a 2/3's chance to win the tournament.

best wishes,
Mason

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Surely this is only if A & B are the now the only players remaining at the point A doubles up. If there's 200 people in a tournament, player A doubles up on player C on the first hand you can't tell me he now has a 67% chance to win outright.

Re: Poker Tournament Formula Revisited
 

[ QUOTE ]
[ QUOTE ]
Hi BigAlK:

Yes. If Player A at some point in a tournament has more chips than Player B then he is a favorite at that instant over Player B. And in your example, Player A does have a 2/3's chance to win the tournament.

best wishes,
Mason

[/ QUOTE ]

Surely this is only if A & B are the now the only players remaining at the point A doubles up. If there's 200 people in a tournament, player A doubles up on player C on the first hand you can't tell me he now has a 67% chance to win outright.

[/ QUOTE ]

Of course not. The example was for a heads-up situation.

Re: Poker Tournament Formula Revisited
 

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Hi BigAlK:

Yes. If Player A at some point in a tournament has more chips than Player B then he is a favorite at that instant over Player B. And in your example, Player A does have a 2/3's chance to win the tournament.

best wishes,
Mason

[/ QUOTE ]

Surely this is only if A & B are the now the only players remaining at the point A doubles up. If there's 200 people in a tournament, player A doubles up on player C on the first hand you can't tell me he now has a 67% chance to win outright.

[/ QUOTE ]

Of course not. The example was for a heads-up situation.

[/ QUOTE ]

The phrase "at some point in a tournament" does not imply we have reached heads-up play.

Re: Poker Tournament Formula Revisited
 

Since we're parsing sentences. Mason's first sentence says that if at some point player A has more chips than player B that player A has more chance to win than player B. The 2nd sentence reads "in your example." Go back thru the thread starting with this one and you'll see that the 2nd sentence is talking about heads up.

Re: Poker Tournament Formula Revisited
 

Thanks for the clarification.

Re: Poker Tournament Formula Revisited
 

[ QUOTE ]
Hey, Mason. Have you given the book a rating yet?

[/ QUOTE ]

I think in the October forum magazine

Re: Poker Tournament Formula Revisited
 

"I thought that most agreed that it is best to play with a 'win at all costs' type mentality because it's higher EV than just playing a weaker game designed to barely get you into the top 10% to make it into the money."

I thought that there may be short-stack situations near the bubble where the best EV is NOT to play to win at all costs?