Poker Advantage and the Gambler's Fallacy

[Zorkion]

Ok mates, I know I haven't posted in a while, but I'm fairly boozed, so listen up. I made a little simulation in excel to demonstrate the effect of an overall advantage percentage in a game like poker.

The excel file is available here: https://dl.dropbox.com/u/8752375/PokerAdvantage.xlsx

Now go download it because this is somewhat interactive.

Ok, as far as using the file goes, here's the idea. in the box to the right of the "ad=" one, you can input a certain % advantage (or disadvantage) that you will have in this theoretical scenario. (eg., for a 1% ad, when you flip a coin, you get 1.005$ for heads, and -.995$ for tails) Start val represents your beginning bankroll, final is your final. min and max show your min and max bankrolls over 100k hands. Its pretty self explanatory.

To refresh the chart, just enter a value of '1' into a blank cell anywhere and hit enter, excel will do the rest.

What does this have to do with poker.

Ok, well heres the general idea. I'll give you some of my observations from playing with this a bit on my own, please feel free to add anything you find.

1) The Gambler's Fallacy.

If you have 0% advantage, you aren't expecting to have have 0 net profit 95% of the time. You're basically expecting to be net winner or net loser about equally.

Linear regressions of the data are important here! You'll find that R-squared values are pretty low in two cases:

a: %advantage is low, or zero.
b: the number of "hands" is low

THIS IS ALL INTUITIVE. Small sample size is effectively USELESS in analyzing your advantage. why? statistics. need more explanation? google it.

The big problem I see with the 0% advantage scenario is that it could be very easy in an actual poker setting to interpret the results as "oh i did well here, but then i got bad at poker, but then i got good again in the end".

The moral of the story is that the slope of your linear regression of your results is going to be the best indicator you have of your advantage in poker. Even so, over 100k "hands" you shouldn't expect it to be great, but take a look at your R-squared value for a general idea of how accurate its going to be. The closer to 1, the more accurate.


2) The more your advantage (or disadvantage) deviates from 0, the more accurate your results are going to be.

I haven't gone through to see what sort of sample size would be needed to get well-corresponding slopes to your % advantage based on the magnitude of your advantage, but i assume its going to be cool and something like the graph of y=(ax)^2, with a being your ad%. Somebody should play with it and post results.


3) Why the heck dont poker tracking softwares give you a linear regression with an r-squared value for your results!?!!

From playing with the simulation, I've found that in general, if the r-squared value of your linear regression is greater than about .9, you can generally expect that your slope is going to be a good indicator of your advantage percentage. Hurray!

But if its not, even like .84, you might be anywhere between +/- 1.5% advantage.



Things I'm interested in because of this:

1) Anyone who can implement linear regression and r-squared values into poker tracking software.

2) A general function for the probability density of r-squared values as a funciton of the magnitude of your advantage% and the sample size.

3) ???????
4) PROFIT.


Things I've learned practically from this:

1) My poker garf is pretty bumpy, so Im probably not as hot as I think I am at poker.

2) Without 100k+ hands, any conclusions I draw from my garfs is pretty much useless, provided that my garfs are bumpy. (point 1 included.)

3) If I was really REALLY good at poker ( or really really bad) my garfs wouldn't be very bumpy. Big surprise here: on average, I'm probably mediocre for my level.

[Zorkion]

oh, I should add: the ad% basically represents your total (current) long term advantage over your competition, accounting for all short-term deviations (tilt, running hot, etc) Its an idealized quantity that doesnt take into account long-term growth.

[xptboy]

not quite sure what you're trying to do but does it have anything to do with variance and standard deviations related to bb/100?

If so there's already a handy calculator for that online: ev++ Poker Tools :: Poker Variance Simulator

It's basically a program that uses statistics tools to figure out how well or how bad you can run over as many hands as you want considering how swingy the type of player you are and your true winrate etc over a long set of hands. There's also some other nifty tools on there like bankroll rate of ruin calculator's etc

[MadMojoMonkey]

It sounds like you know how to do linear regression. So, while the software isn't doing it, you can export your bankroll data to excel and do it yourself.

If all you want is the graph, then you don't even have to do any calculations. Just make a column of your BR, and plot it, add the best-fit line w/ data... or use =LINEST() to get the standard deviations on the slope and intercept (handy info).

I don't use any tracking software, but I do enter my end-of-day BR into an excel spreadsheet and make pretty graphs. On my monthly graph, I plot "hands/100" on the x-axis and "profit/bb" on the y-axis. This means the slope of the linear regression is my winrate (in bb/100) for the month. Oh, I also force the best-fit line to have y-intercept at 0, as that's where my profit started at the beginning of the month.

*note: is there nothing in the tracking software that estimates your overall winrate at a given stakes? I'm surprised by that. It's just too easy a number to calculate and what poker player isn't interested in their personal, overall EV?

[Zorkion]

well yes, it shows you your overall winrate, but as far as I can tell, thats just based on one point of data: the most recent level of your bankroll. As opposed to the whole set of data. and it certainly gives you no r-squared values or measures of the accuracy of your winrate

[MadMojoMonkey]

well yes, it shows you your overall winrate, but as far as I can tell, thats just based on one point of data: the most recent level of your bankroll. As opposed to the whole set of data. and it certainly gives you no r-squared values or measures of the accuracy of your winrate

This is an incorrect assessment of the way Excel calculates best-fit lines. If you plot your data on a graph, you can add a best-fit line. When you do so, you have the option to display both the equation and the r^2 value on the graph. (The graph in your download has this stuff, so maybe I misunderstand you.) As long as you've used a straight line as your best-fit model, you have a "normal" linear regression. Whatever model you use, it's a least-squares regression, which is what you learned in statistics class.

The =LINEST() command on excel will give the m and b parameters for linear regression equation y = mx + b, as well as the r^2 value and the s^2 values (which are the variance of the m and b parameters)

This is the same algorithm that Excel uses for the best-fit line in the graph, but you can display the info in cells, instead of a text box in the graph.

So, since you know your starting bankroll, you can set the y-intercept to that value, then you can use LINEST to tell you the slope and s^2 value, meaning your winrate is
m +/- SQRT(s^2)
which tells you the uncertainty in your slope (winrate).

[seven-deuce]

Hey MMM,

What do i plug in where to get the line?

=LINEST(known_y's,[known_x's],[const],[stats])

[MadMojoMonkey]

OK, so column A is just an index column, so {1;2;3;...}
column B is your monies (or profit) after each hand (or day or whatever).

known_y's is column B
known_x's is column A (since it is just an index, excel will assume this if there is no input here)
[const] can be TRUE or FALSE, with TRUE being normal, and FALSE omitting the y-intercept (setting b = 0).
[stats] needs to be set as TRUE if you want the full output.

*note: Excel considers TRUE equivalent to 1 and FALSE equivalent to 0.

IMPORTANT:
BEFORE you put in the formula, you need to select a 5 row by 2 column array of cells for the output. So, off to the side of your data, you select a 5 tall by 2 wide set of cells.
With ALL of the cells selected type
=LINEST(known_y's,,0,1)
Notice I omit the known_x's with 2 commas in a row, and use 0 for false, 1 for true. The zero makes the y-intercept 0.
SUPER IMPORTANT:
hold Ctrl and Shift when you press Enter

All 10 cells should be filled with numbers now:

The top left is the slope
and the cell beneath that is the +/- for the slope. *note: you do not need to take the SQRT of this, I was mistaken in my prior post.
3rd down on the left is the r^2 value
next is the F-statistic
and the bottom left is the regression Sum of Squares .

The top right is the intercept
and the cell beneath that is the +/- on the intercept. (again, no SQRT required).
3rd down on the right is the standard deviation of the y-values.
next is the degrees of freedom
and on the bottom right is the residual Sum of Squares.

EDIT: This will not make a line, this will tell you the equation for the line and the uncertainty in the slope and intercept.
If you want to see the line, just lasso columns A and B and select insert->scatter
This will make a plot of the data, and you can right-click one of the data points in the plot and select "add trendline". In the dialogue box, near the bottom there are check-boxes for "Display equation on chart" and "Display r^2 value on chart"

you can't get the uncertainty in the slope and intercept without LINEST, though.

[Zorkion]

this is a small point, but I'm not convinced that it makes sense to set the y-intercept to zero. can you explain your rationale behind doing this as opposed to not?

[MadMojoMonkey]

I force the y-intercept to zero in my monthly plots because I'm estimating my monthly winrate, which is based on profit. My profit at the beginning of the month is 0, and this is a known value, so I do not need to estimate it.

In my main graph, I have BR on the y-axis and # of hands on the x-axis, so my BR didn't start at 0. Again, I have it forced the linear regression to start at the correct value. Again, this is a known value.

Honestly, in my main graph, the linear regression only changes a very small amount if I do not force the intercept to the actual value. I guess this means that the slope is more reliable than if the intercept moved dramatically, but really, that's why I'm calculating the standard deviation of the slope in the first place.

Admittedly, I do not know if the closeness of the modeled y-intercept to the actual y-intercept is directly correlated with the standard deviation of the slope of the linear regression. That part of statistics - the part where we use statistics on our statistics so we can put error bars on our error bars - I got lost on that stuff.

[eugmac]

nice bump? it says, in japanese, something like "storage bags not included". That's some f-in weird spam, man.

[MadMojoMonkey]

I wonder, does anybody ever use this technique for estimating their winrate?