Testing ICM -- some questions for discussion

[John Paul]

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Good questions. Thanks for joining the discussion.

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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.

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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.


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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

[marv]

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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.


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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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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

[the shadow]

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:

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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?


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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

[marv]

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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?


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(The following assumes the proof I gave is OK and we play with a randomized button.)

Yes, it's linear: at the moment the 3rd place player is decided, if one of the remaining two players has 100x% of the chips and both are playing optimally, his equity is 0.3 + 0.2x .

If the players are using optimal strategies they'll never make -CEV plays, even if their opponent were suddenly to deviate. If one of them deviates to the point where he does make a -CEV play worth -100x% chips, his oppo gains 0.2x equity.

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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?


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I interpret this as: if I lose the tourney I lose $10, if I win I get $11, but we still start with equal numbers of tourney chips. Then with optimal play from both players, if I have 100x% of the chips at some point, my equity is 21x-10.

In general one should only ever make -CEV plays if you think it will induce the opponent to make even worse -CEV plays (in total) in later hands of the tournament. Of course a play may be -CEV against one type of oppo while +CEV against another.
Optimal plays will always be >=0 CEV against any oppo.

Marv