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  #11  
Old 11-08-2005, 04:08 AM
ZeeJustin ZeeJustin is offline
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Default Re: The Coinflip Game!

I just want to point out that my "proof" doesn't even factor in tournament payout considerations. What I mean is, my proof would be more accurate in winner take all tournaments. Having a flatter payout structure would only decrease the solution (66k) and reinforce Mason's conjecture further.

This means that both payout structure and number of players will affect the rate at which each chip gives you equity.
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  #12  
Old 11-08-2005, 04:36 AM
CardSharpCook CardSharpCook is offline
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Default Re: Conjecture and Question

1. Your equity has an upper bound of the 1st place prize. Your expectation cannot increase linearily when there is an upper bound. I am too far out of 9th grade math, but this is called an S-curve? Your equity cannot increase to infinity.

2. First place isn't worth the sum of all the chips: Win you win the 1000 man tourney, you will have 10,000,000 chips. However, 1st place is a mere 25% or $2.5M. Even though chips were issued at a 1chip = $1 scale, they don't pay that way.

3. Part of your equity comes from your seat, part of it comes from your chips. The part that comes from your seat does not change at the same rate as your chip stack. Seat eq is based on #of players left, chip eq is based on %of total chips.

4. Part of your equity comes from the opportunity present at the table. If you have 20K chips, but no one else at your table has more than 10K, 10K of your chips have no opportunity attached to them. The shortest stack at the table has the most opportunity equity.

5. You are expected to quadruple your buy-in. While it is uncommon to start with a double up, it is, to some degree expected. You have a 4X edge because you find ways to double up. Your equity does go up (obviously) but because you're expected to increase your chip stack, it cannot double. If you were expected to half your buy in, doubling your chips would have a greater effect? This doesn't sound right.
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  #13  
Old 11-08-2005, 04:53 AM
Pat Southern Pat Southern is offline
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Default Re: Conjecture and Question

[ QUOTE ]
5. If you were expected to half your buy in, doubling your chips would have a greater effect? This doesn't sound right.

[/ QUOTE ]

This is something that not many of us have thought of because we like to consider ourselves winning poker players, but I think that this may help find some conclusions. Are you saying that somebody with a $4999 expectation can have an expectation over $10000 after doubling, or somebody with a $5001 expectation still will not have an equity over $10000 after doubling?
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  #14  
Old 11-08-2005, 05:30 AM
ZeeJustin ZeeJustin is offline
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Default Re: Conjecture and Question

[ QUOTE ]
[ QUOTE ]
5. If you were expected to half your buy in, doubling your chips would have a greater effect? This doesn't sound right.

[/ QUOTE ]

This is something that not many of us have thought of because we like to consider ourselves winning poker players, but I think that this may help find some conclusions. Are you saying that somebody with a $4999 expectation can have an expectation over $10000 after doubling, or somebody with a $5001 expectation still will not have an equity over $10000 after doubling?

[/ QUOTE ]

The amount of skill you have changes the rate of equity change when you double. See my example if you want to try to apply some math to it. Change the 60% figure to 30% or 80% and see how the outcome changes.

Or, if you're too lazy to do the math, someone with max equity (guarenteed first place) will gain zero from a double up.
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  #15  
Old 11-08-2005, 05:46 AM
A_PLUS A_PLUS is offline
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Default Re: Conjecture and Question

The really simple answer is that our skill edge doesnt multiply along with our chips.

Basically, we calculate EV by (for an average player)

The sum of:
% of total chips * payout for 1st place
% of total chips * payout for 2nd place
.....

= Our Expected Value of playing the tournament


So, something else has to be at play here for a better player.

They calculate their EV by:

% of total chips + SKILL ADJUSTMENT * 1st place payout
% of total chips + SKILL ADJUSTMENT * 2st place payout
..........

So for Mason's example, the player has a skill advantage of
X. Which is just how much more on average they expect to finish in certain spots.

So, when we double our chips, we double the portion of the above calculation that is from % of total chips. If we do not also double the "SKILL ADVANTAGE" Factor, the EV does not double along with it.

Intuitively, this makes sense. The further we progress through a tournament, the more our % of total chips effects the outcome. If we did in fact try to double or "skill adjustment" along with total chips, we would quickly reach an unreasonable EV.

So, it comes down to the decay of the skill adjustment factor. It is mathematically impossible for it to double, but if it were to remain constant, It quicky becomes unimportant in decesion making.

Actually, that doesnt feel very different from how many pros play. The edge they require before risking all of their chips getting smaller and smaller as the field condenses, converging to anything greater than 0, at a point.


That being said, I find it hard to believe that many people have a 4x edge in a large MTT, and 99% of the players on this board are going to use this as an excuse to play weak tight
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  #16  
Old 11-08-2005, 06:19 AM
Jman28 Jman28 is offline
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Default Re: Conjecture and Question

Okay, I'll try the other side since nobody else has, and because it's what I believe. First let me state my conjecture: In the situation Mason described, your expectation probably does not double if your chip count does. However there are times when it can double and more. Therefore, Mason's conjecture that it definitely does not double is incorrect.

Basically, I think that the SKILL ADJUSTMENT factor in A_PLUS's equation can change value when your chips stack does. I don't think anyone will argue that having a bigger stack allows you to make more +EV plays, especially on the bubble.

If the game is such that the gain you get from having a big stack at bubble time (or at other times) is sufficiently large, your expectation can double (or even triple, theoretically) when you double up. This gain may be different depending on tourney structure, or the tendencies of your opponents.


As an extreme example, let's say we're 25 handed, with 20 spots paying. Blinds 100/200.

You have 6000 chips and everyone else's stack is between 3500 and 4500.

Everyone is trying to make the money desperately. So desperately, in fact, that if the pot is raised in front of them by someone who covers them, they will fold any hand but AA. If they cover the raiser, they will play a much more 'standard' game.

Of course this is a huge exaggeration of most real players' tendancies, but think of the massive advantage you gain in having 6000 chips compared to having 3000. This hopefully shows that a much smaller but very real advantage exists when you have a big stack in more realistic scenarios. This advantage may sometimes be enough to counter the effects of the standard equity assumptions most of us have.

Edit: 2 more points.

Your edge doesn't stop when the bubble bursts either, as people are attempting to climb the prize ladder and will still fear you.

If what I'm saying is true, the implications would be strange and counter intuative. In a situation where doubling up would more than double your equity, it would be +EV for two players next to each other to decide to go all in blind the first time it was folded around to them in the blinds. This makes me doubt my conclusion a little bit, but I'd like to hear what others have to say.
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  #17  
Old 11-08-2005, 07:59 AM
Oatmeal Oatmeal is offline
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Default Re: Conjecture and Question

***Feel free to disregard this post. I know it is rambling, but I believe I have a better chance of learning something here for having engaged the problem.***

Bravo, Shane, and for what it's worth (not much I'm sure)I agree, but in this hypothetical I am... "a great tournament player. Perhaps one of the best."

With this as the given my instincts tell me doubling on the 1st hand certainly couldn't have more than doubled my expectation and it is probably a bit less than double due to my believing that each chip I win is worth less than the one before.

A guesstimate would put my new expectation between 72500 and 77500, but that is just a feeling which I'm sure doesn't help with Mason's question all that much. Really interested in seeing some more of these replies.

-sloth
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  #18  
Old 11-08-2005, 08:12 AM
betgo betgo is offline
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Default Re: Conjecture and Question

I agree with this. Your expectation less than doubles.

However, in most situations where you have an average to twice average stack from the point where half of the field has been eliminated onward, doubling up results in much more than a doubling of your expected payout, regardless of your skill level.
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  #19  
Old 11-08-2005, 08:18 AM
Spee Spee is offline
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Default Re: Conjecture and Question

[ QUOTE ]
So if my conjecture is correct, I would like to hear good reasons why this is the case. If it's not correct, I want to hear those reasons as well.

This should make for a good discussion and thanks in advance for the help.

[/ QUOTE ]

Hello Mason,

With respect to the start of the tournament, IMHO, it is intuitively obvious that your conjecture is correct. Typical tournament payouts are more or less linear whereas doubling up is exponential. At some point, the maximum expectation will be reached regardless of the amount of doubling. Stack size will increase at a faster rate than expectation.

With respect to latter stages of the tournament, I don't think this is so much the case. For example, does doubling up further increase expectation once into the money? I would think that it most definitely would (or is that stating the obvious??).

When I was thinking about your conjecture, the immediate analogy that popped into mind was Tiger Woods in golf. After all, he is the greatest tournament player of the current generation, if not all time.

For 2005, he played in 21 PGA events and made 13 top 10s and 6 wins. As an aside note, he also dogged it and missed the cut in 2 events.

So let's say Tiger is roughly 2:1 or 2.5:1 to win the event (as he usually is rated by the bookies here in the UK). He then promptly goes out and eagles the first hole to take a two shot lead on the field. Does that increase his expectation? Maybe a little but not too much. He is after all quite a bit better than even money to finish in the top 10. But it is also very early in the tournament.

Now lets say it is Saturday afternoon, and he has already made the cut (i.e., in the money) and is now tied with 9 other players for the lead. These 10 players together are 4 shots clear of the rest of the field. Now Tiger goes out and eagles the first hole to take a two shot lead over his 9 closest competitors and 6 on the rest of the field. Does that increase his expectation? I would say yes, proably by a lot.

So now back to poker, the greatest tournament player is now already somewhere in the money. Does a double increase his expectation more so than at the beginning of the tournament? Yes, I think intuitively that it would have to. Or maybe this is restating the obvious as well??

Just my own wrong opinion FWIW ...

Cheers,
Spee
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  #20  
Old 11-08-2005, 08:52 AM
Ansky Ansky is offline
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Default Re: Conjecture and Question

lots of golf analogies on this forum lately.
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