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  #91  
Old 05-14-2005, 08:18 PM
John Paul John Paul is offline
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Default Shadow + Marv = ? or: What is the source of betterness?

Hello,
I have been following this thread over the past few days. If I understand everything, Marv has showen that if both players push every turn, then their % equity =% chips. Shadow has pointed out that a the meaning of being a "better" player than your oppenent is that your %equity > %chips.

Taken together - It might be worth thinking about how betterness occurs. If the short stack is small enough - their blind will put them all in. At this point, for both players %equity = %chips. So neither can play better here.

Even if the blinds do not put the short stack all in, it sems to be the 2+2 dogma that you can have a stack so short that you have to push any 2 cards, and the big stack should always call. In this case neither player can play better, just worse than pushing. Can this stack size be determined analytically in terms of stack sizes and blind sizes?

So, if you are going to be better than someone HU, that is have your $Equity>Chip Equity, you must have to do it when the stacks are more equal in size. That is, there has to be some fixed chip level where pushing/calling pushes is not the best possible play. But I wonder if you could devise an optimum strategy based on the equity of individual hands vs. random hands. I don't think that you could necessarily come up with an analytical solution for every situation, but I suspect that you could narrow down the possiblities.

What do folks think?

John Paul
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  #92  
Old 05-14-2005, 09:13 PM
John Paul John Paul is offline
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Default Oops, Sorry Gumpzilla

Sorry, It was gumpzilla who made the little drawing of the $ equity vs. chip equity for a player who is better than their opponent heads up. Just want to give credit where it is due.
JP
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  #93  
Old 05-15-2005, 01:08 PM
the shadow the shadow is offline
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Default Re: Testing ICM -- some questions for discussion

I've given more thought to gumpzilla's chart of an equity function where the hero is more skilled (or playing a better strategy) than the villian. Even though the artistry leaves a little to be desired [img]/images/graemlins/smile.gif[/img], I think gumpzilla's chart is a correct way of looking at it.

I approached the problem by using the gambler's ruin formula. Here's the hypothetical -- hero and villian start with a stack of 500 chips apiece for a total of 1,000 in play. For each play, hero and villian bet $1 on the flip of a coin. If it comes up heads, hero wins $2. If it's tails, villian wins $2. Using that scenario, if we know hero's stack size, can we figure out his equity?

Now, if the coin is perfectly balanced, then the equity function is linear. The hero's equity equals his or her percentage of the chips in play.

But what if the coin just slightly favors the hero? I assumed that the coin was weighted so that it falls heads 50.1% of the time. Using the gambler's ruin formula, I then calculated the hero's equity for the following stack sizes:

Hero Equity
0 0.00
10 0.04
50 0.18
100 0.34
200 0.56
300 0.71
400 0.81
500 0.88
600 0.93
700 0.96
800 0.98
900 0.99
1000 1.00

Here's a chart of the equity function using these assumptions:



As you can tell, the two charts, using different assumptions, are both concave.

That means that gumpzilla's statement

[ QUOTE ]
Also interesting is that since f(A) + f(B) = 1, this means that B grows more slowly than linear for small stacks and faster than linear for big stacks (I think, correct me if I'm wrong here.)

[/ QUOTE ]

is correct. If the hero starts with a small stack, his equity grows fastest when he's short-stacked. For example, as the hero's stack grows from 100 to 200 chips, his equity grows from about 1/3 to over 1/2.

(Another interesting aspect of this chart is that a minor skill disparity of only 50.1% can have major effects when multiplied over 1000s of coin flips. That may illustrate why a push-or-fold strategy at high blinds may take away much of an opponent's skill advantage. If an unskilled player can force the tournament to come down to a single flip of a coin, he's more likely to win then if the tournament is decided by repeated coin flips at which a minor disadvantage is multiplied time and time again.)

Given that gumpzilla and marv may have put to bed any debate about the shape of an equity function for equally matched players each using an optimal strategy, I'm inclined to agree with gumpzilla's suggestion that developing an equity function for skill disparities may be more interesting.

The Shadow
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  #94  
Old 05-15-2005, 01:30 PM
eastbay eastbay is offline
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Default Re: Testing ICM -- some questions for discussion

[ QUOTE ]

Given that gumpzilla and marv may have put to bed any debate about the shape of an equity function for equally matched players

[/ QUOTE ]

Let's not get confused, now. We know how to construct examples for equally matched players who have strongly nonlinear equity functions.

Marv's argument is based on optimal play only.

eastbay
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  #95  
Old 05-15-2005, 01:30 PM
the shadow the shadow is offline
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Default Re: Shadow + Marv = ? or: What is the source of betterness?

Good questions. Thanks for joining the discussion.

[ QUOTE ]
So, if you are going to be better than someone HU, that is have your $Equity>Chip Equity, you must have to do it when the stacks are more equal in size.

[/ QUOTE ]

John Paul, I'm either not understanding what you're saying or I think you may be mistaken.

If the hero is more skilled than the villian, then the equity of the hero's chips is greater than the hero's chips as a percentage of all chips in play for all levels of the hero's stack between 0 and 100%. In other words, the hero's skill advantage will manifest itself at any stack size and may be more powerful at smaller stack sizes.


[ QUOTE ]
Even if the blinds do not put the short stack all in, it sems to be the 2+2 dogma that you can have a stack so short that you have to push any 2 cards, and the big stack should always call. In this case neither player can play better, just worse than pushing. Can this stack size be determined analytically in terms of stack sizes and blind sizes?

[/ QUOTE ]

If you haven't done so already, take a look at the heads up and hand ranking threads in the favorite threads list. A lot of work already has been done to calculate push top x% and call with top y% of hands at different blind levels. poincaraux also has posted some similar analysis on the poker stove website.

The Shadow
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  #96  
Old 05-15-2005, 01:32 PM
the shadow the shadow is offline
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Default Re: Testing ICM -- some questions for discussion

Good clarification. I agree.
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  #97  
Old 05-15-2005, 01:57 PM
pergesu pergesu is offline
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Default Re: Testing ICM -- some questions for discussion

[ QUOTE ]
[ QUOTE ]
As a lawyer, I'm continually surprised by how much information you can get by simply asking. Assume that an online site were willing to assist. What and how much information would you be looking for?

[/ QUOTE ]

The size of the blinds at the time the match gets to be HU seems like it would be a relevant factor.

One issue that interests me about the data collection is how many data points one should take from each tournament. Let's say I play 20 hands heads up against my opponent; should we use this as 20 data points? My gut feeling is no, we shouldn't, because any deviations from average behavior become magnified.

As an oversimplified, unrealistic example of what I'm talking about, say I get heads up with blinds of 100, my stack being 2000, villain's stack being 8000. Now let's say he's ridiculously tight, so tight, in fact, that he'll fold anything but aces. Now I should win this just about all of the time, and so I'm going to skew the data horrendously if you take multiple data points from this tournament that show my small stack overcoming a major disadvantage and winning.

[/ QUOTE ]

I haven't read all the posts yet, I'll do that when I have more time, so I don't know if you guys came to a conclusion on how many data points to use.

It seems to me that you could just use the final hand of each tourney. There are only 4k possible starting stacks in an 8k chip tourney, and 5k possibilities in a 10k chip tourney. So if you managed to get 5 million final hands, that ought to give you sufficient data. Then you can examine the deviation between stack% and win%. I don't know how useful that would be in coming up with a new chip modelling theory, but it could be used to validate the linear theory that Sklansky suggests.

Again, this may have been brought up already, and perhaps even shot down thoroughly. Just a thought I had while reading through the first page.
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  #98  
Old 05-15-2005, 07:37 PM
John Paul John Paul is offline
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Default Re: Shadow + Marv = ? or: What is the source of betterness?

[ QUOTE ]
Good questions. Thanks for joining the discussion.

[ QUOTE ]
So, if you are going to be better than someone HU, that is have your $Equity>Chip Equity, you must have to do it when the stacks are more equal in size.

[/ QUOTE ]

John Paul, I'm either not understanding what you're saying or I think you may be mistaken.

If the hero is more skilled than the villian, then the equity of the hero's chips is greater than the hero's chips as a percentage of all chips in play for all levels of the hero's stack between 0 and 100%. In other words, the hero's skill advantage will manifest itself at any stack size and may be more powerful at smaller stack sizes.


[/ QUOTE ]

I don't think you can be a better player when one stack is really small HU, although in a way it is a trivial result. If I only have 50 chips, and the blinds are 100/200 then both me and my opponent are going all in next hand (and the one after that if I win) and there is no possibility to show any skill. If an always push strategy results in %equity=%chips, then for someone to show skill, it has to be at a time when the blinds don't force one player all in HU. Like I said, this is pretty trivial, but if I am following the debate, that would imply that equity relationship is linear at the extremes. This would not depend on any characteristic of the 2 players, as the blinds dictate their strategies. However, I may mis-understand the debate here.

Thanks for the links. I have been playing limit ring games for a few months, but I am pretty new to SnG's so I am still catching up to the rest of the class. I hope folks keep pursuing these things both analytically and empirically. Perhaps there will be some results that help folks play, and it is interesting in its own right anyway.

John Paul
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  #99  
Old 05-15-2005, 08:52 PM
marv marv is offline
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Default Re: Shadow + Marv = ? or: What is the source of betterness?

[ QUOTE ]
[ QUOTE ]

If the hero is more skilled than the villian, then the equity of the hero's chips is greater than the hero's chips as a percentage of all chips in play for all levels of the hero's stack between 0 and 100%. In other words, the hero's skill advantage will manifest itself at any stack size and may be more powerful at smaller stack sizes.


[/ QUOTE ]

I don't think you can be a better player when one stack is really small HU, although in a way it is a trivial result. If I only have 50 chips, and the blinds are 100/200 then both me and my opponent are going all in next hand (and the one after that if I win) and there is no possibility to show any skill. If an always push strategy results in %equity=%chips, then for someone to show skill, it has to be at a time when the blinds don't force one player all in HU. Like I said, this is pretty trivial, but if I am following the debate, that would imply that equity relationship is linear at the extremes. This would not depend on any characteristic of the 2 players, as the blinds dictate their strategies. However, I may mis-understand the debate here.

Thanks for the links. I have been playing limit ring games for a few months, but I am pretty new to SnG's so I am still catching up to the rest of the class. I hope folks keep pursuing these things both analytically and empirically. Perhaps there will be some results that help folks play, and it is interesting in its own right anyway.

John Paul

[/ QUOTE ]

Here's a thought:

For small stack sizes, certainly we'd expect our players' difference in skill level to be smaller as there will be less preflop play. At the extreme, as people have noted, once one of you is blinded all-in, then skill has left the building, you just call.

But if the short stack doubles up a few times, he may now still have less equity than the linear equity function would suggest if he tends plays a short stack poorly, so his equity vs chips graph is linear in the left hand corner but its angle is much less that 45%. This might show up in the data - that short stack players play worse than they should.

One thing I learned from my tussle with eastbay is that if you can play each hand in a +cev way, you're certain to get at least the linear equity. In fact your additional equity is exactly the expectation of the sum of the 'cev edge' you have over your opponent over the remaining hands.

I'd like to know if shortstacked players (when there are 3+ players) do (or should) take -cev actions? That can't be too hard to test?


Marv
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  #100  
Old 05-16-2005, 10:14 AM
the shadow the shadow is offline
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Default Re: Gumpzilla + Marv = ? or: What is the source of betterness?

Marv, thanks for offering the proposed proof of a linear equity function for a HU match where each player uses an optimal strategy. I'm still thinking about it and am looking forward to eastbay's comments.

This thread started with a suggestion for some empirical work, but due to the contributions of you, gumpzilla, eastbay and others, has made more progress on the analysis side. To that end, you write:

[ QUOTE ]
I'd like to know if shortstacked players (when there are 3+ players) do (or should) take -cev actions? That can't be too hard to test?


[/ QUOTE ]

Well, if you're counting me, shortstacked players certainly take -CEV actions, but not because it's the right thing to do. [img]/images/graemlins/crazy.gif[/img] Seriously, it's a good question. I'll give it some thought.

In the meantime, it seems to me that there're still some productive questions to explore about heads up play. In addition to incorporating skill differences, here're two perhaps easier ones to start with:

Assume that we're down to the last two players of a SNG. The payout is 0.5 to 1st place and 0.3 to 2d place. Each player uses an optimal strategy. Is the equity function still linear? What does it look like? Should one player make -CEV plays?

Assume that one player gives another player odds in a heads up match. Again, each player uses an optimal strategy. Is the equity function still linear? What does it look like? Should one player make -CEV plays?

The Shadow
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