Tag Archives: statistics

The Benford’s Law Hustle™

Posted by on March 9, 2012 Comments Off on The Benford’s Law Hustle™

Hi, poker blog readers! I was going to post this on my new nerd blog, and maybe I’ll do that as well. But, I figured that poker players might like this, even if it’s not strictly related to poker.

Basically, this is the simplest, most convincing +EV prop bet that I know of. In other words, it’s an awesome hustle. I’ve occasionally imagined walking into a poker room, finding the nearest guy with a visible newspaper or smart phone, and leaving with his money in my pocket, but unfortunately, nerds don’t make good hustlers.

Here’s how you do the Benford’s Law Hustle™:

  1. Walk into a casino and find one of a very large class of (pseudo)random number generators that roughly satisfy Benford’s law. (Don’t cheat and google that yet.) You can pick random numbers from newspaper articles (Turn to page A5 and find the first number in the first article) or something similar. Tons of things will work; just about the only things that won’t will be numbers from casino games. (E.g., dice and keno boards are no good.)
  2. Find someone who’s willing to bet on the first digit of the random number. (The first digit of 2,458,193 is two. Nothing fancier than that)
  3. Bet on one and two.
  4. Give your opponent both eight and nine.
  5. Lay two to one… (E.g., you pay him $200 if the number is 8,283 or 9,722, and he pays you $100 if the number is 10,136 or 2. If it’s 637, then no money changes hands.)
  6. Make an absurd profit.

Wait… what?

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Life as a HU SnG Pro by the Numbers (It’s Awesome… If You’re Good and Have Rakeback)

Posted by on March 31, 2011 8 comments

In previous posts, I looked at how variance affects players who play large-field MTTs, smaller-field MTTs, NLHE 6-max cash, and 6-man and 9-man STTs, and I found quite a bit of difference. Next up: HU SnGs. (Still to come: HU NLHE, FR NLHE, HU PLO, 6-max PLO, HU LHE, 6-max LHE, and FR LHE. I’m going to be quite busy for at least the next couple weeks, so please don’t hold your breath. Follow me on twitter if you’d like to know when these posts go up.)

HU SnGs are quite simple statistically, so I will make exactly one assumption: rake that is 1/22 of the rake-free buy-in. This is the standard rake for turbos on Stars and FTP at most stakes (e.g. $110+$5), so everything that I say will be exact for those games, but if you play games with a different rake:buy-in ratio, the numbers will be a bit different. To get most of the results, I won’t even bother to assume a normal distribution; I’ll just use the binomial distribution, which is an exact statistical representation of a HU SnG. (This won’t actually change the numbers at all after rounding, but it just requires a bit of extra algebra from me, and it’ll appease some of the statistical purists in the audience.)

Anyway, with that out of the way, let’s jump in. Say you’re a HU SnG pro with a solid ROI of 3% at the Stars $115s. You’re also moonlight as the world’s greatest physicist, and you invent a cloning machine that you planned to use to resurrect Einstein, Lincoln, Ghandi, etc., thus ushering in a new era of global age of peace and prosperity. But, then you remember that rent’s due soon, so instead you make 99,999 clones of yourself and get to grinding. You figure 1k HU SnGs each should cover rent plus some standard expenses. (Trips to the Rhino get expensive when there are 100,000 of you…)

How does this HU clone army fare? Good question!

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Life as an STT Pro by the Numbers (It’s a Lot Better Than You Probably Think)

Posted by on March 20, 2011 10 comments

In previous posts, I looked at how variance affects players who play large-field MTTs, smaller-field MTTs, and NLHE 6-max cash. Now, I thought I’d grab some low-hanging fruit in the form of sit-n-gos. It turns out that 9-handed and 6-handed STTs are very similar statistically, so I’ll lump them together below (I justify this in my assumptions section). HU SnGs are next in line, and should be done in a day or two (no promises).

If you’re a fellow nerd, you might want to read about [slider title=”my assumptions”]

  1. I’m only going to consider Poker Stars $114 9-mans and $119 6-mans. Some sites have different payout structures. In particular, some sites spread 10-mans and/or 5-mans instead, which obviously changes the payout structure and changes the numbers as well. This analysis will still give a decent picture for all roughly similar games, but keep in mind that it is explicitly an analysis of the Stars $114s and $119s.
  2. I’m going to assume normality. STTs are pretty close to normal over samples of 100+ tourneys and essentially indistinguishable over 500+ tourneys, so that shouldn’t be a problem. This follows directly from properties of the binomial distribution.
  3. I’m going to assume constant standard deviations. In theory, standard deviation for an STT player is dependent on her win distribution. So, players with different ROIs can be expected to have different standard deviations, and even players with the same ROI could have different standard deviations. In practice, these effects are tiny: Standard deviations vary by only about 20% in 9-man STTs and only about 10% in 6-max STTs over reasonable finish distributions for serious players. I’m not looking to estimate confidence intervals within 10 or 20%, so this should be fine.
  4. I’m going lump 9-man and 6-man SnGs together, with a 1.5 BIs/tourney standard deviation for both 6-max and 9-man STTs. I didn’t initially plan on doing this, but it turns out that the numbers are almost identical for the two. Typical standard deviations are about 0.05 BIs/tourney higher for 9-man STTs and about 0.05 BIs/tourney lower for 6-man. So obviously this approximation is good enough for my purposes.

(Of course, if you know basic statistics, everything in this post is derivable easily from the above. So, the real meat of this post is contained in the assumptions, which are all justified by a bit of behind-the-scenes research with sharkscope and windows calculator and some discussions with friends of mine. The rest is essentially just watching me divide by SQRT(n) and plug in to my favorite z-score calculator repeatedly)[/slider].

In my previous posts, I didn’t consider rakeback because it typically varies by stake and becomes much less important at the higher stakes. STT players make a large percentage of their income from rakeback, VIP programs, and bonuses, even at the highest stakes, and it doesn’t vary much by stakes. So in this post, I’m going to talk about “effective ROI”, not ROI. Effective ROI is a phrase (that I made up) that means your ROI after you consider rakeback, bonuses, VIP rewards, etc. In other words,

[latex] \displaystyle (\mathrm{Effective\ ROI}) = (\mathrm{Raw\ ROI}) + \frac{(\mathrm{Rakeback\ etc.}) }{(\mathrm{Buyins,\ including\ all\ rake)}} [/latex]

 

With that housekeeping out of the way, let’s look at the numbers. Take a player with an effective ROI of 7%. (He might, for example, play $114s with a raw ROI of 4% and earn an additional 3% from Stars VIP program and bonuses–equivalent to a RB % of about 38%.) What happens if he and his 99,999 identical grinder twins (!) play 1,000 STTs each? Well… this does:

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Life as a NLHE 6-max Cash Game Pro by the Numbers (It Ain’t Too Bad)

Posted by on March 7, 2011 24 comments

(Update 3/8: I originally completely forgot to talk about what happens with different winrates. I added this analysis to the bottom of the post.)

In two of my previous posts (1 2), I crunched some numbers to show that the variance in large-field MTTs is pretty damn crazy. Now I’m going to turn my attention to NLHE 6-max cash games. (Sorry for the delay. I could claim to have been busy, but mostly I’ve just been lazy.)

This is actually pretty easy thanks to the statistician’s best friend, the central limit theorem. For sample sizes of at least a few thousand hands, you can just take your standard deviation and winrate and use the normal approximation. (For sample sizes of less than a few thousand hands…. Who cares?) This makes the math really easy, and in a way, it’s responsible for the relatively cushy lifestyle of cash players–The normal distribution is a lot cleaner than the distribution that I found for MTT players. (Proof that the distribution is normal)

(Another consequence of this mathematical convenience is the fact that some other people have done this analysis already. While my posts on MTTs were, as far as I know, the first honest attempt at tackling that problem correctly, this post will mostly just explain what’s already known to the nerdy contingent of the poker world and anyone else with a basic understanding of statistics. I’m just bothering to share this information with our less nerdy brethren in a way that I feel is reasonably clear. To that end, I’ll mostly just leave out any explanations, but suffice it to say that all of this comes from very basic facts about the normal distribution. However, I think everybody who plays poker should be able to do some basic statistics, so I might make a tutorial explaining where these numbers come from at some point.)

To see how that works out in practice, let’s look at a basic example. Take a solid 6-max NLHE grinder with a 5 bb/100 win rate (I’m going to use bb/100, not PTBB/100 in this post. An unfortunate tradition ported from LHE leads many people–including PTR, PT3, and sometimes me–to call a BB or a PTBB twice a bb.) with a fairly typical standard deviation of 90 bb/100. What happens if 10,000 clones of this guy play 50,000 hands each? Well, this does:

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Life as an Online MTT Pro II: The Numbers Are Back, and They’re Out for Blood

Posted by on February 3, 2011 21 comments

(This is a follow-up to this post. You probably want to read that first if you haven’t already.)

My previous post about the variance in MTTs got some interest and plenty of criticism, so I thought I’d follow up with a (slightly) less sloppy post that considered some criticism. In future posts, I plan to look at various forms of cash games, STTs, and DoNs. (If you’d like to share your data for any of these things, please get in contact with me.)

First of all, I should clarify something: Some people are definitely better off playing large-field MTTs than other games. I’ll make an argument below that almost no small/mid-stakes players fit into this category in the current climate. For mid/high-stakes player: If you’re insanely good at MTTs and you’re down to put in lots of volume, then play them. If you really enjoy large-field MTTs and you’re willing and able to deal with their ridiculous variance, then play them. If you’ve got a really sweet backing deal, then play them. If for whatever other reason you think you’d like to play them, I won’t pretend to know what’s best for you. But, make sure you seriously consider the extreme variance that’s involved. Before this, I haven’t seen anyone present this information correctly. Thus these posts. (I have seen people use the normal approximation to look into this, but as I showed here, that method doesn’t actually work.)

Anyway, there’s lots of fun stuff that I didn’t do with Shaun’s data that I’d like to do with my shiny new data set (see the section at the bottom for more about this sexy piece of data). For example, varying buy-in sizes obviously increases variance. For Shaun’s data, I looked at buy-ins between $55 and $216, i.e. with a max buy-in about 4x his min buy-in. But what if we don’t vary the buy-ins at all? Here’s the data for that:

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Life as an Online MTT Pro by the Numbers (It’s Hard)

Posted by on January 27, 2011 23 comments

(After you read this post, you might want to check out my follow-up to it here.)

I tell a lot of people not to play large-field online MTTs for a living. I’ve always thought that the variance is just way too high for most professionals to trust their livelihood (and sanity) to large-field MTTs instead of cash, smaller-field MTTs, or STTs. But, admittedly, I’ve given this advice without any direct evidence to back it up. I’ve been meaning for a while to see what the numbers say, and this post will be a tentative first step.

Ideally, what I’d like to do is do a nice controlled study where I pick a few representative players based on past results and use their results over the next few months as my data. (Alternatively, I could take the results of one of the large backing groups. If anyone who backs 20+ people would be down to share some information, let me know.) But, that requires more motivation than I’ve been able to muster, so I decided to do a much rougher study: I grabbed Shaun “SFD” Deeb’s tourney results from OPR (with Shaun’s permission) and played around for a few hours. Here’s what I found:

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