Quote:
Originally Posted by EuG00
I'll try to clarify again with an example:
Let's pretend that we have 9 imaginary identical players playing the same exact format SNG over and over.
1st gets 65%, 2nd 35% of the total prize pool
In the short term, obviously some will be bigger winners than others due to luck variance. (since skill is identical it's not part of this equation)
The are different variance factors, like:- Waking up with AA and AK pre more than others
- or getting AA to KK more than others
- or winning coin flips more than others
- or winning AI in general (coin flips, suck outs, best hand holding up)
- or flopping flushes
- or hitting set over set
- or maybe flopping TP, etc. etc. etc.
How would your rank these different variance factors (or others I didn't mention) in order of importance?
Now the 2nd part, obviously after a large number of games everybody will be at around 0 (even)
I know it won't actually be exactly 0, but let's just say it's some absurdly small margin of error away from 0.
What do you think this number is? 100 games, 1,000, 10,000, 1,000,000?
Thanks!
The second part can be calculated and shown, the first part you can't say how a certain hand influences the final result of the game.
For the 2nd question I'll start by assuming that we are only really concerned with 500 games or more and then it is quite reasonable to assume the results will fall as though from a Normal/Gaussian distribution. (To calcluate it for less games it might be better to use the maths of another distribution, perhaps the Poisson distribution but the more games the closer and closer we get to Gaussian - and for poker 500 games isn't much).
I'll change the numbers to avoid some messy decimals, so 10 players and 60%, 40% all exactly equal in ability. This would be the same as a 10 slot roulette wheel with each slot acting as a win or second for a player and instead of using one ball we chuck in two at once - gold gets first, silver gets second.
If we play blocks of 1000 games over and over and plot out the results for these we would find that the average for each player would mark out a very nice 'bell curve'.
We know a lot about this bell curve like how often you get within a certain amount and it works out that roughly 68% of the time a player would get a result betwenn + or - of 1 standard deviation of the mean (in this case all players equal, no rake, the mean is zero).
This bell curve tails off to -infinity and +infinity but to the human eye it looks like a bell of width 6 standard deviations wide (only a very few results ever come out further than 3 sd's from the mean, 99.7% are withing this).
So if we can calculate the mean we can work out how wide our results will appear to be - as the number of games is increased the bell gets taller and sharper and this, in a way, is what we mean by the results converging.
To work out the sd of a number of games we work out the variance of a single game and mutliply by the number of games to get the variance for this number. Then we take the sqrt of this and this is the sd for this number of games.
10 players all pay 1 and winner gets back 6.0 so wins +5.0, 2nd gets +3.0, losers lose 1.0
Mean = 0.0 (we know this is due to each player having equal ability, so no win in the long run)
Variance of an individual win = (0.0 - 5.0)^2 = +25
Variance of an individual 2nd= (0.0 - 3.0)^2 = +9
Variance of an individual loss= (0.0 -(-1.0))^2 = +1
We have to weight these for how often they happen to get the variance per game (this is basically what jukofyouk showed earlier).
Variance per game = (0.1 * 25) + (0.1 * 9) + (0.8 * 1) = 4.2
The var per game is 4.2
Var per 1000 games = 4.2 * 1000
Sd of 1000 games = Sqrt(4.2 * 1000) = 65
So for 1000 games we would expect a 'bell' curve to look to the human eye 6 sd's wide, so from -195 to +195 wide but in percentage terms compared to the total we wagered (1000) this is 195/1000. So 19.5-% to +19.5%.
If we play 4000 games we have
Var per 4000 games = 4.2 * 4000
Sd of 4000 games = Sqrt(4.2 * 4000) = 130
So now this bell curve "looks" 6 x 130 wide, ie. from -390 to + 390, but again in percentage terms when we wagered 4000 this is -390/4000 and +390/4000 = -9.75% and + 9.75%
If you notice the 4000 game percentage range is exactly half that of the 1000 game one. And this is because the amount wagered is just going up by 'n' the number of games but the sd is only going up by Sqrt(n) a smaller factor.
Every time we take a block 4 times the original size the percentage range is narrowed by half and this is how our results will be converging.
(warning, I may have ballsed some of this up but I think the method should be fine)
Last edited by BaseMetal2; 10-24-2014 at 10:29 AM.