Something I\'ve been thinking about
 

OK, a few questions. Lets for arguements sake say that although all these plays are 0 EV (ie, ignore pot size as well), you want to do all of them. Also assume, that he has every hand in his range with equal possibility. The question is, what 1 would you prefer, or does it make no difference....and why?

Example 1: Villain has 2 hands in his range, One you have 0% equity against, the other, you have 100% equity against.

Example 2: Villain has 3 hands in his range, 2 of them have 75% equity vs your hand, 1 of them has 0% equity vs your hand

Example 3: Villain has 3 hands in his range, 2 of them have 25% equity vs your hand, 1 of them has 100% equity vs your hand

Example 4: Villain has 2 hands in his range, One you have 25% equity against, the other, you have 75% equity against.

Example 5: Villain has 10 hands in has range. 5 Have 0% equity vs your hand and 5 have 100% equity vs your hand.

Re: Something I\'ve been thinking about
 

Hand reading for our sake matters and villain having to make more difficult decisions, matters also. Perhaps lower variance also matters, but it's a little difficult to take all of this into consideration theoretically.

Re: Something I\'ve been thinking about
 

the only thing i would care about here is the information i got from a call. I don't care about variance. So I can't really answer, you'de have to be alot more specific.

Re: Something I\'ve been thinking about
 

Example 1= Example 5 unless i'm reading this wrong.

Re: Something I\'ve been thinking about
 

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Example 1= Example 5 unless i'm reading this wrong.

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That's what I thought too.

Re: Something I\'ve been thinking about
 

i think what matters as to which of the 5 examples i want will depend very heavily on villain.

Re: Something I\'ve been thinking about
 

Assuming all the pots are the same size all of these are 0 EV. I don't think I'd care.

Re: Something I\'ve been thinking about
 

I'm cnfused.

Re: Something I\'ve been thinking about
 

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I'm cnfused.

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HINT: think varience

Re: Something I\'ve been thinking about
 

i strongly prefer less variance. you guys should too, because it means you can move up faster. which is highly +EV in a meta-sense. not to mention everyone tilts, even those of you who think you don't, and reducing variance reduces tilt.

if you're getting at something deeper than that, you lost us all.

Re: Something I\'ve been thinking about
 

a seperate thought, for those where you have narrowed his range down to 2 or 3 hands, you're more likely to see if you were right. if his range is 10 hands, its less likely that he will have a hand that is in his range that you didn't think was in his range.

Re: Something I\'ve been thinking about
 

Since the distribution is the same and the overall equity is the same, my first impression is that the variance will actually be the same for all of these situations. That is, a coin flip is a coin flip, no matter how complicated the "coin" is. In all cases, we have a sample size of a single hand, which we will win or lose, where we have 50% equity. There is no difference.

What am I missing?

Re: Something I\'ve been thinking about
 

To those more mathematically inclined than me-

Am i right in thinking that Example 4 has the least variance?

Also, in terms of variance, 2=3, and 1=5 right?

Re: Something I\'ve been thinking about
 

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Since the distribution is the same and the overall equity is the same, my first impression is that the variance will actually be the same for all of these situations. That is, a coin flip is a coin flip, no matter how complicated the "coin" is. In all cases, we have a sample size of a single hand, which we will win or lose, where we have 50% equity. There is no difference.

What am I missing?

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There is no lottery-style variance with 100% equity and 0% equity.

Re: Something I\'ve been thinking about
 

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Since the distribution is the same...

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What do you mean by that? Distributions of those 5 examples are not the same...

Re: Something I\'ve been thinking about
 

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There is no lottery-style variance with 100% equity and 0% equity.

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This is not true. 50% of the time we'll have 100% equity, and 50% of the time we'll have 0% equity. This is the same as having 50% equity 100% of the time.

Re: Something I\'ve been thinking about
 

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Since the distribution is the same...

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What do you mean by that? Distributions of those 5 examples are not the same...

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Hero has 50% equity against villain's stated range in all 5 examples.

When we know the villain's range with 100% certainty, the only thing that matters is our equity against that range. The composition of that range is irrelevant.

The composition of villain's range generally DOES matter, because we aren't 100% confident in that range. We need to analyze, even if only by gut feel, how sensitive our equity is to small changes in the range.

Re: Something I\'ve been thinking about
 

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There is no lottery-style variance with 100% equity and 0% equity.

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This is not true. 50% of the time we'll have 100% equity, and 50% of the time we'll have 0% equity. This is the same as having 50% equity 100% of the time.

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WHAT?!?!

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Example 1: Villain has 2 hands in his range, One you have 0% equity against, the other, you have 100% equity against.


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This means that the lottery-variance of cards that have yet to come have 0 influence on the outcome.

Re: Something I\'ve been thinking about
 

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This means that the lottery-variance of cards that have yet to come have 0 influence on the outcome.

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No. I'm talking about equity, not outcome.

Re: Something I\'ve been thinking about
 

in a vaccuum i don't care, but in reality it depends on the villain

Re: Something I\'ve been thinking about
 

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Since the distribution is the same...

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What do you mean by that? Distributions of those 5 examples are not the same...

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Hero has 50% equity against villain's stated range in all 5 examples.

When we know the villain's range with 100% certainty, the only thing that matters is our equity against that range. The composition of that range is irrelevant.

The composition of villain's range generally DOES matter, because we aren't 100% confident in that range. We need to analyze, even if only by gut feel, how sensitive our equity is to small changes in the range.

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Yea, ignore my comment, I am hallucinating. Same mean, same distribution, same variance.

Re: Something I\'ve been thinking about
 

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Since the distribution is the same and the overall equity is the same, my first impression is that the variance will actually be the same for all of these situations. That is, a coin flip is a coin flip, no matter how complicated the "coin" is. In all cases, we have a sample size of a single hand, which we will win or lose, where we have 50% equity. There is no difference.

What am I missing?

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say our equity vs. his range is 50%. however say we both have the same hand w/ no redraws. our equity is and always is regardless of what card comes, 50%. this means no variance since we're always chopping the pot. if his hand range includes two hands one w/ 0% and one w/ 100% there will be variance because the results are different.

Re: Something I\'ve been thinking about
 

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i think what matters as to which of the 5 examples i want will depend very heavily on villain, the action to the point in the hand when the equity is being evaluated, and the way the equity will develop over future streets, if there are any.

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At least as far as deciding how to play the hand, and how profitably you can play the hand.

Generally, I think I would prefer a situation where there are a variety of equities, most of the equities are relatively close to 50%, stack/pot ratio is high, and I will be better able to evaluate the effect of future cards on the equities than villain (which may be a function of my hand, his range, our relative skill, or a combination).

If you're simply asking this as a variance question, I don't think it should matter.

Re: Something I\'ve been thinking about
 

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Lets for arguements sake say that although all these plays are 0 EV (ie, ignore pot size as well), you want to do all of them.

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What plays? Is this an all-in situation?

I'm going to assume that we're moving in here, and the villain is calling with one of the 5 ranges you listed (is this what you had in mind?).

If so, it doesn't matter which range he has - both the expectation and variance are the same in all 5 cases.

Re: Something I\'ve been thinking about
 

This is only a varience question.

My question is, are the varience in all 5 cases (and in particular, the ones I've asked) the same?

Re: Something I\'ve been thinking about
 

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This is only a varience question.

My question is, are the varience in all 5 cases (and in particular, the ones I've asked) the same?

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It's very, very close to the same, yes. As suggested above, there is a slight reduction in variance for the non-binary cases due to chopped pots. In practical terms, that's a small enough difference to be negligible, however.

A coin-flip is a coin-flip, no matter how the problem is broken down. That's a large part of the beauty of equity calculations. They abstract away the details of individual hands and give us a broader understanding of what's going on.

Re: Something I\'ve been thinking about
 

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My question is, are the varience in all 5 cases (and in particular, the ones I've asked) the same?

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Yes.

Re: Something I\'ve been thinking about
 

I never took stats but intuitively I'd say that they go 1,2,3,4 from lowest to highest variance. Also, 2 and 3 have the same variance. My reasoning is that you can calculate the variance of each individual event and then sum using their likelyhood. So 1 has 0 variance. 2 and 3 are same since the case of 25% equity vs 75% equity are the same since you sway the same amount from the average (ie 75 or 0, 25 or 100).

Re: Something I\'ve been thinking about
 

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My question is, are the varience in all 5 cases (and in particular, the ones I've asked) the same?

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Yes.

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Isn't 1 just like taking insurance ala hellmuth on high stakes poker? So there is 0 variance

Re: Something I\'ve been thinking about
 

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A coin-flip is a coin-flip

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Say you buy a lottery ticket $2, and you have 1/10000 to win $19 998. It's the only price. This situation is a coinflip but it has a lot more variance than a head-or-tail coinflip.

Re: Something I\'ve been thinking about
 

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A coin-flip is a coin-flip, no matter how the problem is broken down. That's a large part of the beauty of equity calculations. They abstract away the details of individual hands and give us a broader understanding of what's going on.

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Ok, but does being all-in with 90% equity have higher variance than being all-in with 50% equity? I think that's the crux of the OP

Re: Something I\'ve been thinking about
 

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Ok, but does being all-in with 90% equity have higher variance than being all-in with 50% equity? I think that's the crux of the OP

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At the time the decision is made, we're never all-in with 90% equity. That's the key. When we decide what to do, we have 50% equity.

Re: Something I\'ve been thinking about
 

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Say you buy a lottery ticket $2, and you have 1/10000 to win $19 998. It's the only price. This situation is a coinflip but it has a lot more variance than a head-or-tail coinflip.

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Being laid the right odds to take a bet and equity are not the same thing. In your lottery example, you do not have 50% equity, even though the bet is break even.

Re: Something I\'ve been thinking about
 

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So 1 has 0 variance.

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No.

Once we are called, it's true that the outcome is certain, but we don't know what hand he has. From our perspective, we win the pot 1/2 the time and lose the pot 1/2 the time.

Thus, we're flipping a coin for the pot, and this will be true for any range against which we have 50% equity. So the distribution of outcomes is the same in all 5 cases, and hence the variance must also be the same.

Re: Something I\'ve been thinking about
 

Ok look, mathematically, if I'm right, examples 2 and 3 have the same variance, which are the 2 situations where variance is higher while number 4 comes 2nd while 1 and 5 have no variance.

E(X) = SUM(px)
V(X) = E(X^2)-(E(X))^2

For 1 : E(X) = 0.5 V(X) = 0
For 2 : E(X) = 0.5 V(X) = 0.125
For 3 : E(X) = 0.5 V(X) = 0.125
For 4 : E(X) = 0.5 V(X) = 0.0625
For 5 : E(X) = 0.5 V(X) = 0

Re: Something I\'ve been thinking about
 

[ QUOTE ]
Ok look, mathematically, if I'm right, examples 2 and 3 have the same variance, which are the 2 situations where variance is higher while number 4 comes 2nd while 1 and 5 have no variance.

E(X) = SUM(px)
V(X) = E(X^2)-(E(X))^2

For 1 : E(X) = 0.5 V(X) = 0
For 2 : E(X) = 0.5 V(X) = 0.125
For 3 : E(X) = 0.5 V(X) = 0.125
For 4 : E(X) = 0.5 V(X) = 0.0625
For 5 : E(X) = 0.5 V(X) = 0

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That would be true if our decision point were after we knew which actual hand villain holds. Since we don't, we can't break it down like that.

That is, the variance is dependent on the amount of uncertainty at the time the decision is made. The underlying composition of that uncertainty doesn't matter.

Re: Something I\'ve been thinking about
 

[ QUOTE ]
Ok look, mathematically, if I'm right, examples 2 and 3 have the same variance, which are the 2 situations where variance is higher while number 4 comes 2nd while 1 and 5 have no variance.

E(X) = SUM(px)
V(X) = E(X^2)-(E(X))^2

For 1 : E(X) = 0.5 V(X) = 0
For 2 : E(X) = 0.5 V(X) = 0.125
For 3 : E(X) = 0.5 V(X) = 0.125
For 4 : E(X) = 0.5 V(X) = 0.0625
For 5 : E(X) = 0.5 V(X) = 0

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If you use E(X) = SUM(px), then you should be using: V(X)=SUM(x-E(X))^2*px.

Edit: and how the hell did you get E(X)=0.5? [img]/images/graemlins/confused.gif[/img]

Re: Something I\'ve been thinking about
 

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Ok look, mathematically, if I'm right, examples 2 and 3 have the same variance, which are the 2 situations where variance is higher while number 4 comes 2nd while 1 and 5 have no variance.

E(X) = SUM(px)
V(X) = E(X^2)-(E(X))^2

For 1 : E(X) = 0.5 V(X) = 0
For 2 : E(X) = 0.5 V(X) = 0.125
For 3 : E(X) = 0.5 V(X) = 0.125
For 4 : E(X) = 0.5 V(X) = 0.0625
For 5 : E(X) = 0.5 V(X) = 0

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That would be true if our decision point were after we knew which actual hand villain holds. Since we don't, we can't break it down like that.

That is, the variance is dependent on the amount of uncertainty at the time the decision is made. The underlying composition of that uncertainty doesn't matter.

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I honestly didn't understand what you said but while writing my answer I found what was wrong in my reasoning. Assuming it's an all-in situation, half the time we're winning, half the time we're loosing, in every situation, it's that simple. You're right imo.

Re: Something I\'ve been thinking about
 

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half the time we're winning, half the time we're loosing, in every situation, it's that simple.

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Exactly. [img]/images/graemlins/smile.gif[/img]

Re: Something I\'ve been thinking about
 

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Edit: and how the hell did you get E(X)=0.5? [img]/images/graemlins/confused.gif[/img]

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sit 2 : E(X) = SUM(px) = (2/3)*(0.75) + (1/3)(0) = 0.5

Aint this right?