empirical equity study

[parappa]

Thanks not only for your work, but also for this quite lucid explanation of what you're on about. As someone with only small amounts of college math, I can pretty easily follow what you're saying thanks to the explanation.

[hansarnic]

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Don't think I understand Eastbay.

Strategy A leads to the player with the most chips at the start of the HU match winning virtually 100% of the time, no?

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Not virtually - exactly.


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Yes, except where starting BB size > stack size, i.e. where small stack cannot fold his BB, wins showdown & becomes big stack

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So that player has more skill (or rather the lower stacked player has no skill as he just folds every hand giving the game to his opponent).


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No. Both players are playing identical strategies. Clearly then both strategies must have equal skill - because they are the same.


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But in reality they are not playing the same strategy. One stack is folding 100% of the time the other is pushing 100% of the time. The smaller stack's strategy will always lead to him losing. It's a bit like saying to both players - when you are sitting in the 1 seat you should push and when you're sitting in the two seat you should fold. This is 'the same' strategy but the actual skill level is pre-determined by events outside the players control (i.e. seat number or starting stack size).

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Strategy B takes skill entierly out of the equation.

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Any two identical strategies takes skill out of the game.


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Not in the examples above

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And yes, there are a couple of proofs of $EV = Chip count in HU in TPFAP, one of which is calculated on the basis of both players going all-in every hand.

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Right, but there is also a 'purely logical proof.' Can you quote it for us? It seems clearly wrong, at the very least through insufficient qualification.

eastbay

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p 105 "There is however a more elegant,purely logical proof that equal players in a symetrical situation must win exactly in proportion to the size of their stacks. Supose these two players agreed to play the same freezout everyday for ten years. Given they are equal players, their long term reults would necessarily be to break even. Well in order for them to break even, the smaller stack must win exactly the same fraction of times as the fraction of the total chips he possesses. This reasoning extends to any number of players as long as they play equally well, and there is no positional advantage for one compared to the other."

[eastbay]

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But in reality they are not playing the same strategy.

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I think you have a strange notion of strategy, then. A strategy is a set of rules for how to act according to context, i.e., if I have AA I am going to limp and move in on any flop. If I have AK and less than 10BB, I am going to move in. Any such set of rules constitutes a strategy. The idea that a strategy must be stack independent is absurd - that's bad poker. In my example, each player is playing according to exactly the same set of rules.

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One stack is folding 100% of the time the other is pushing 100% of the time.

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In the same context, they make the same move. That's playing the same strategy.

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The smaller stack's strategy will always lead to him losing. It's a bit like saying to both players - when you are sitting in the 1 seat you should push and when you're sitting in the two seat you should fold. This is 'the same' strategy but the actual skill level is pre-determined by events outside the players control (i.e. seat number or starting stack size).


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There's a big difference between your example and mine, and that is that a reasonable poker strategy will never be absolute seat number dependent - there's never any advantage to considering that as a variable in a strategy, but a reasonable poker strategy WILL be dependent on relative stack size - there's advantage to be gained (or lost if misapplied) by considering relative stack size. So it is reasonable to consider the space of stack-dependent strategies, but it is not reasonable to consider the space absolute seat number dependent strategies.

eastbay

[eastbay]

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p 105 "There is however a more elegant,purely logical proof that equal players in a symetrical situation must win exactly in proportion to the size of their stacks. Supose these two players agreed to play the same freezout everyday for ten years. Given they are equal players, their long term reults would necessarily be to break even. Well in order for them to break even, the smaller stack must win exactly the same fraction of times as the fraction of the total chips he possesses. This reasoning extends to any number of players as long as they play equally well, and there is no positional advantage for one compared to the other."

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Sklansky either means something strange by "equal players" or "in a symmetrical situation", or "the same freezout" or he's simply wrong.

The two players break even in my scenario, and they do not win in proportion to the size of their stacks. He says "the same reasoning" except that he supplies none.

There is only one strict symmetry requirement for the equity function and that is that a player must win half the time if he has half the chips. Any curve which passes through (0,0), (0.5,0.5), and (1,1), and has the property that eq(x) +eq(1-x) = 1. is admissible. This includes an infinite number of variously skewed "S" curves, ranging from fully linear to a step function for the degenerate case I provided.

eastbay

[jcm4ccc]

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it's not clear to me now if there's a problem in using the same tournament results multiple times so long as the chip stack distribution for which you're recording it doesn't occur more than once in the same tournament

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It is wrong to use the same tournament results multiple times in your analysis, regardless of the chip stack distribution. Here’s an example of why it’s wrong:

Suppose you have played in 2 tournaments in a $100 tourney. When it got down to 4-handed, you had 2000 chips (both times). According to ICM, you should win 20% of the prize pool in each tournament. $1000 * 2 * 20% = $400 you are expected to win.

Let’s say that you win one tournament, and you lose one tournament. Your total prize is $500, or 25% of the prize pool. The ICM model predicted you would win 20% of the prize pool, and you won 25%. You did better than expected. This is pretty straightforward.

Now, let’s suppose that the bubble lasted 4 hands in the tournament that you won, and the bubble lasted 1 hand in the tournament that you lose. You decide to include each hand in your analysis (as long as the chip stack distribution is different). Here’s what your data might look like (I don’t know how to do tables):

Tourney__Hand__Chip Count__ICM Pred__Actual
1________1_____2000________20%_______50%
1________2_____2100________21%_______50%
1________3_____1900________19%_______50%
1________4_____1500________15%_______50%
2________1_____2000________20%_______00%
TOTAL_____________________19%_______40%


By this analysis, it looks like you won 40% of the prize pool, when actually you only won 25% of the prize pool. You look a lot better than you actually are. That is because you overweighted the tournament in which you won.

We can reverse this, of course. Let’s suppose that the bubble lasted 4 hands in the tournament that you lost, and the bubble lasted 1 hand in the tournament that you won. Here’s the analysis:

Tourney__Hand__Chip Count__ICM Pred__Actual
1________1_____2000________20%_______00%
1________2_____2100________21%_______00%
1________3_____1900________19%_______00%
1________4_____1500________15%_______00%
2________1_____2000________20%_______50%
TOTAL_____________________19%_______10%


Now you look worse than you actually are. That is because you overweighted the tournament in which you lost. You need to weight each tournament equally, or your analysis doesn’t work.

[eastbay]

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it's not clear to me now if there's a problem in using the same tournament results multiple times so long as the chip stack distribution for which you're recording it doesn't occur more than once in the same tournament

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It is wrong to use the same tournament results multiple times in your analysis, regardless of the chip stack distribution. Here’s an example of why it’s wrong:

Suppose you have played in 2 tournaments in a $100 tourney. When it got down to 4-handed, you had 2000 chips (both times). According to ICM, you should win 20% of the prize pool in each tournament. $1000 * 2 * 20% = $400 you are expected to win.


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I think you're missing something. My equity is not only a function of my stack, it's a function of the remaining stacks as well. If I have 2000 chips and the chips are evenly distributed amongst the other players, clearly I have less equity than if for my same 2000 chips, one other player only has 1 chip left. This is what I mean by chip stack distribution.

My question relates to using data from the same tournament to compute the expectation for _different_ chip stack distributions. Your example was lumping all the data into one equity expectation. I am asking about the case where you are computing equity _for each distribution_.

eastbay

[jcm4ccc]

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it's not clear to me now if there's a problem in using the same tournament results multiple times so long as the chip stack distribution for which you're recording it doesn't occur more than once in the same tournament

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It is wrong to use the same tournament results multiple times in your analysis, regardless of the chip stack distribution. Here’s an example of why it’s wrong:

Suppose you have played in 2 tournaments in a $100 tourney. When it got down to 4-handed, you had 2000 chips (both times). According to ICM, you should win 20% of the prize pool in each tournament. $1000 * 2 * 20% = $400 you are expected to win.


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I think you're missing something. My equity is not only a function of my stack, it's a function of the remaining stacks as well. If I have 2000 chips and the chips are evenly distributed amongst the other players, clearly I have less equity than if for my same 2000 chips, one other player only has 1 chip left. This is what I mean by chip stack distribution.

My question relates to using data from the same tournament to compute the expectation for _different_ chip stack distributions. Your example was lumping all the data into one equity expectation. I am asking about the case where you are computing equity _for each distribution_.

eastbay


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Plain and simple, your "actual equity" in these charts is wrong if you are using more than one data point from the same tournaments, for the reasons I outlined above.

stacks: actual equity, icm predicted equity, difference, standard error of sample, number of data points in this range
0,0,0,0: 0.282925; icm = 0.239223; delta = 0.0437015; std err=0.00508585; N=1265
0,0,0,1: 0.266667; icm = 0.195756; delta = 0.0709111; std err=0.0189215; N=90
0,0,1,0: 0.244944; icm = 0.203231; delta = 0.0417128; std err=0.0190025; N=89
0,1,0,0: 0.215385; icm = 0.18758; delta = 0.0278049; std err=0.0148834; N=117
1,0,0,0: 0.426316; icm = 0.393979; delta = 0.0323365; std err=0.0221329; N=19

Actually, I can't see how the "actual equity" could ever be carried out to six decimal places. In any tournament, you win either 0.5, 0.3, 0.2, or 0.0 of the prize money. No matter how you add these numbers, you would need at most one decimal place. How do you ever manage to win 28.2925% of the prize pool?

[jcm4ccc]

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Actually, I can't see how the "actual equity" could ever be carried out to six decimal places. In any tournament, you win either 0.5, 0.3, 0.2, or 0.0 of the prize money. No matter how you add these numbers, you would need at most one decimal place. How do you ever manage to win 28.2925% of the prize pool?

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Actually, that's a stupid statement on my part. Please ignore.

[eastbay]

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it's not clear to me now if there's a problem in using the same tournament results multiple times so long as the chip stack distribution for which you're recording it doesn't occur more than once in the same tournament

[/ QUOTE ]

It is wrong to use the same tournament results multiple times in your analysis, regardless of the chip stack distribution. Here’s an example of why it’s wrong:

Suppose you have played in 2 tournaments in a $100 tourney. When it got down to 4-handed, you had 2000 chips (both times). According to ICM, you should win 20% of the prize pool in each tournament. $1000 * 2 * 20% = $400 you are expected to win.


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I think you're missing something. My equity is not only a function of my stack, it's a function of the remaining stacks as well. If I have 2000 chips and the chips are evenly distributed amongst the other players, clearly I have less equity than if for my same 2000 chips, one other player only has 1 chip left. This is what I mean by chip stack distribution.

My question relates to using data from the same tournament to compute the expectation for _different_ chip stack distributions. Your example was lumping all the data into one equity expectation. I am asking about the case where you are computing equity _for each distribution_.

eastbay


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Plain and simple, your "actual equity" in these charts is wrong if you are using more than one data point from the same tournaments, for the reasons I outlined above.


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You're missing the point. I know this because the reasons you gave are not relevant to the question at hand. In your example you lumped all of the data together. My initial results did this too, and I agree that this is wrong for the reasons you gave.

But now there's a new question, which I've stated above but which I think you aren't understanding yet. Do you understand what ICM is? I'm not saying that to be condescending, you've made some statements which seem to indicate that you don't.

eastbay

[jcm4ccc]

I understand ICM.

It's your study. I'll step aside for now.