Conditional Prob: Estimating % of Turn Raise Bluffs

[gaming_mouse]

We want to estimate the % of HU turn raise bluffs in the game we play based on hand histories. For convenience, "showdown" means that the oppo who raised the turn ended up showing his hand down, and "bluff/r" means "bluff raise."

P(showdown & bluff/r) = P(bluff/r | showdown)*P(showdown)

Both quantities on the right can be estimated. That is, if need be, we can manually look at all showndown hands in which there was a turn/r and decide if the raiser was bluffing or not, enabling calculation of P(bluff/r | showdown). P(showdown) is just the % of HU hands where the turn raiser shows down.

So we now have an estimate of P(showdown & bluff/r) which will converge to the true value.

Note that:

P(showdown | bluff/r) = P(showdown & bluff/r)*P(bluff/r)

Which means that we can estimate the value we are ultimately interested in -- P(bluff/r) -- as:

P(showdown | bluff/r)/P(showdown & bluff/r)

The problem now is that we have no airtight way of estimating P(showdown | bluff/r). There are some possibilities for crude estimates, however:

1. We could assume that bluff raises are exactly as likely to be called down (and hence shown down) as raises in general (real and bluff combined), and estimate P(showdown | bluff/r) using P(showdown). This seems like a bad idea, as I'd guess bluff raises are, in general, more likely to be called down, though I have no idea HOW much more likely.

2. We could use information about how often OUR OWN bluff raises (or those of other players whose hand histories we have access to) are called down, and use that to estimate overall population stat. This seems like a better option than 1., but is still flawed, since I'd guess that the bluff raises of TAGs are less likely to be called down than the bluff raises of players in general. Intuitively, however, this bias (at least to me) seems like a less egregious error than the bias in 1. In addition, we could estimate the magnitude of the error if we had access to histories for playes with higher VPIPs.

I'm curious if anyone else has other ideas about how to solve these problems, and would like to hear thoughts on the above ideas as well.

TIA,
gm

[jason1990]

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1. We could assume that bluff raises are exactly as likely to be called down (and hence shown down) as raises in general (real and bluff combined), and estimate P(showdown | bluff/r) using P(showdown).

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Do you mean you would estimate P(showdown | bluff/r) using P(showdown | raise)? I like this idea, but I think the former might be slightly smaller. For example, if you call my turn bluff raise, then bet into me on the river, I would probably fold more often than if my original raise has been legit. I don't know though. The formula doesn't distinguish between "good" and "bad" bluffs. In principle, some bluffs may be more transparent than others, so that only "good" bluff raises would be as likely to be called down as legit raises.

The second idea seems a little fishy to me. You want to estimate P(bluff/r from opponent), but in the formula you would use P(showdown | bluff/r from gaming_mouse). Maybe it's not that big of a deal, but the mental red flag popped up when I read that.

Also, I think you're multiplying where you should be dividing, so the final formula is the reciprocal of what you wrote.

As a practical matter, do you know how many hands you would need to gather enough data for this?

By the way, are there sites where you can see the folded hands in the hand history? Or is that just a myth I heard.

[gaming_mouse]

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Do you mean you would estimate P(showdown | bluff/r) using P(showdown | raise)?

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Yes.
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I like this idea, but I think the former might be slightly smaller. For example, if you call my turn bluff raise, then bet into me on the river, I would probably fold more often than if my original raise has been legit. I don't know though. The formula doesn't distinguish between "good" and "bad" bluffs. In principle, some bluffs may be more transparent than others, so that only "good" bluff raises would be as likely to be called down as legit raises.

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Very interesting point, about the bluffer giving up. An opposing force to the point I made about bluffs being more likely to be called down, and I don't know which would be strong. Of course, we could look at the cases of "bluffer has position" and "bluffer OOP" separately to answer your concern.

As to your point of "good" and "bad" bluffs.... My first pass will not distinguish between these two, as this obviously makes the problem much, much more difficult. It is hard enough trying to define a turn raise bluff to begin with (in a way that can be programmed). Finding the criteria for determining what is a "good" and what is "bad" bluff is exponentially harder. In fact, simply getting good players to agree about this in specific cases on the forum is hard.


[ QUOTE ]
The second idea seems a little fishy to me. You want to estimate P(bluff/r from opponent), but in the formula you would use P(showdown | bluff/r from gaming_mouse). Maybe it's not that big of a deal, but the mental red flag popped up when I read that.

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Yes, this is a good point. As I said, you would want to get data from a bunch of players (preferably with diverse styles) for this method to be accurate.


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As a practical matter, do you know how many hands you would need to gather enough data for this?

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I don't. I'd like to have at least a million or so. I'll be able to make better estimates after I start doing a little research.

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By the way, are there sites where you can see the folded hands in the hand history? Or is that just a myth I heard.

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Not that I know of. That would be an amazingly rich source of data though. Let me know if you find one.

Thanks for your comments,
gm