Probability of losing stream

[rachelwxm]

Probability of losing stream (something you might find interesting)

Recently there are lots of discussions regarding losing stream in SNG games, I find it interesting to think about this problem theoretically given my statistical physics background. Generally you could ask two questions:

1. For any N games, what’s the probability that you lose them all assuming your ITM is x?

The answer is simply (1-x)^N according to binomial distribution. For some realistic number ITM=40% loosing stream of 6 games happens 4.6% of time. So if you record 106 games, you should probably see 5 of those on average.

2. Assuming you play long enough, on average how many games do you need to play before seeing a losing stream of N games?

This is not an easy one, more like questions asked by financial companies like D.E.Shaw during interviews. However, induction is the way to go.
Assuming A(N) is the average games you have to hit when you first see a loosing stream of N. then we have
A(N+1)=
(1-x) chance of A(N)+1
x chance of A(N)+1+A(N+1)
so A(N+1)=(1-x)*(A(N)+1)+x*(A(N)+1+A(N+1))
or A(N+1)=(A(N)+1)/(1-x)
so that for ITM40% A(0)=0 A(1)=1.7 A(2)=4.4

I have put this list for anyone who want to verify assuming ITM=40%. We see that losing stream of 20 is very unlikely achievement, 10 is not rare for over a few hundred games.

1 1.666667
2 4.444444
3 9.074074
4 16.79012
5 29.65021
6 51.08368
7 86.80613
8 146.3435
9 245.5726
10 410.9543
11 686.5905
12 1145.984
13 1911.64
14 3187.734
15 5314.556
16 8859.26
17 14767.1
18 24613.5
19 41024.17
20 68375.28

Hope this is interesting and let me know if you find any math problems.
/images/graemlins/smile.gif

[parappa]

This is quite interesting. Thank you for posting it.

[rybones]

I teach english, so I have no sense of the math here. However, if I am reading it correctly (and this is a big assumption), it looks like a 40% itm player will likely see 6 to 10 losses in a row over the course of 100 games?

If this is true, then we have another way to analyze success. Several posts have suggested that 100 games is a small sample from which to evaluate your true potential. However, if you were to look at your itm after 100 games and your losing streaks, the combination might provide some useful information. E.g., say your itm was 48% but you saw two losing streaks of 6 or more games you might fairly say it is not yet time to move up a level. On the other hand, if you saw an itm of 42% but only had one losing streak of 5 games, you should feel more confident about your play. does this sound about right to anyone else?

Ryan

[KJ o]

No, I don't think ITM finishes and losing streaks are independent, so your losing streak numbers really doesn't add anything.

Or rather, it could say something about your game that is independent of ITM finishes, such as "I make a killing on week ends but lose on weekdays" or "I lose after playing for more than 3 hours because I get tired and unfocused".

I think the best way is to use the AM spread sheet and move up when your risk of ruin is sufficently small. What is sufficient is obviously highly individual.

[rachelwxm]

[ QUOTE ]
I teach english, so I have no sense of the math here. However, if I am reading it correctly (and this is a big assumption), it looks like a 40% itm player will likely see 6 to 10 losses in a row over the course of 100 games?

[/ QUOTE ]
yes, for my last 106 games, my worst losing stream is 6 and I have 3 of them. My ITM is somewhat higher than 40%. So it's in line with this analysis. I would like to hear number from others as well. /images/graemlins/smile.gif
[ QUOTE ]

If this is true, then we have another way to analyze success. Several posts have suggested that 100 games is a small sample from which to evaluate your true potential. However, if you were to look at your itm after 100 games and your losing streaks, the combination might provide some useful information. E.g., say your itm was 48% but you saw two losing streaks of 6 or more games you might fairly say it is not yet time to move up a level. On the other hand, if you saw an itm of 42% but only had one losing streak of 5 games, you should feel more confident about your play. does this sound about right to anyone else?


[/ QUOTE ]
Absolutely, using an analogy used in statistical arbitrage I am doing now. We generally use Sharp Ratio (return/std) as a mesure of success, but we also use draw down (max percentage lose in a year) to mesure good strategy. I would be more confident if they do agree with each other.

[rybones]

I agree totally with your analysis here, but I am not sure we are in disagreement. Perhaps the addition of "means your not ready for the next level" focused to much on the end of the analysis. What I really meant was that looking at your losing streaks could provide a means of finding leaks and I think that is what you are saying as well? I was looking at This as yet another way find leaks just as you might use hand analysis, bubble play analysis, or any other type of analysis that effects your itm. As an English teacher you'd think I would have been more clear, sorry.

Ryan

[AleoMagus]

While I agree that your analysis is correct, I think that as far as SNG play is concerned, reality may be a little different than the stats suggest. How is this possible?

I think that for many, the tendency to see big losing streaks (or winning streaks for that matter, which can be calculated in the same way) is actually GREATER than these numbers. The reason for this is that while we have easily calculatable % values that we can attach to things like out ITM% and our ROI%, our 'actual' game-by-game % may fluctuate. For example, we may have a week where we are on vacation and get to sleep in a lot and feel great and whether we realize it or not, we may be going into the game with a higher 'actual' ITM% than our stats say we are. Similarly, whenever we lose a couple in a row, our percentages may drop slightly because we are trying to compensate or may be tilting slightly.

I'm sure that I'm not the only one who can testify to big losing streaks where in the last 2 or 3 losses I know I made some absolutely god awful plays that I'd never make in a good frame of mind. It's rare I imagine, but some players might react in an oppsite fashion whereby they play a higher ITM% game when losing and get overconfident and have it drop when winning.

I've actually thought a bit about this in the past and even thought about getting some sims done whereby winning or losing streak analysis is done on the basis that ROI and ITM goes up by a certain amount after wins and down by a certain amount after losses.

I guess the reason why I think this is the case is just some of the evidence I've observed. I believe that the majority of SNG players I've talked to have somewhat larger streaks than the numbers suggest they should have (both winning and losing).

Am I wrong here and do the stats actually take this into account already? Any Thoughts?

Regards
Brad S

[rachelwxm]

I agree the most important assumption of my analysis is that each game is independent, which is obvious not true in reality. It does not take into account the situation if you are on tilt because of losing stream or drunk or emotionaly distressed so that you are not on your A game or your network went down when you are 4 tabling etc. These all leads to positive autocorrelation of your results. So the reality could be much worse.
However, if you always play at the same level, your results should be close to this analysis. /images/graemlins/cool.gif

[chill888]

Good point, I played sober last night and really struggled.

[Irieguy]

Yeah, Aleo and Rachel, I am really struggling to wrap my mind around this issue. On the one hand, Rachel's stats look perfectly sound (to an unsophisticated statistician), but on the other hand, they don't correlate with what many successful SNG players have seen.

My ITM is very close to 40% at the $55 level, so these numbers apply to me very well. Accordingly, I should have a stream of 14 OOTMs once every 3187 SNGs. Well, I've only played about 1000 and it's happened 3 times.

Now, I understand that in and of itself, that doesn't mean anything. But all of my streams for losses 2-14 in a row (14 is my worst so far, thank god) have occured more frequently than they "should."

Still no big deal, I know, but lots of other successful SNG players seem to have similar experiences. Aleo, your explanation seems to make sense... but it doesn't "feel" right. I don't know.

One other comment: every single time I've moved up in limits (4 times), I've started out with a losing stream of at least 8 OOTMs immediately. When I moved up to the $55's I had a stream of 14 and had to move back down. Then I tried again and had a stream of 12 and decided to just play through it, which fortunately worked out ok. This seems to suggest that when I first move up I play scared or something, and for that little stretch I'm playing poor poker with an average ITM% of much lower than 40%. Also interesting is that after my first 100 SNGs, my ITM and ROI started to very steadily move up. Every 100 SNGs I finish, my overall numbers are better. Maybe I gain confidence and play better? Maybe I'm now playing at an ITM of much better than 40%? The problem is, I still have pretty egregious streams. I almost never go 48 hours without a stream of 8 OOTMs in a row. I'm always shocked to look at my long columns of "(-$55)" and find my overall ITM% to remain pretty stable at 40-42%.

Thank you very much for your work, Rachel, and I'm very interested to hear about what other players have to say on the topic.

Irieguy

[AleoMagus]

I'm actually now pretty convinced that I am right. It "felt" like somehow bad reasoning to me too at first but something rachel said in the response to me made it clear.

SNGs results are not independent of one another.

They should be in theory, I know, but they are not. Tilt, Focus, lack of sleep, computer crashes, drunkenness/images/graemlins/confused.gif, etc...
These things all affect SNG results in groups so that you can expect similar results to cluster slightly

Strange as it sounds, how you do in one SNG does affect how you do in the next. Or at least, factors which affect how you do in one SNG affect how you do in the next.

I suspect if I played 1 SNG every two days for the rest of my life, I'd probably run into streaks just like rachel has shown. My results would be very independent (actually maybe not even then. Someone could die in my family or I could suffer a trauma of my own, etc... These could have very long term effects) but when you play 10 or more in a day, it's easy to let a bad run continue longer than it should.

Regards
Brad S

[chill888]

Just another small point that leads to autocorrelation of nearby results:

Often - especially at non Party sites where less games are starting - you often play clusters of games against similar groups of people. If this cluster is full of great players, you may be more likely to go through a bad patch than usual.

[rachelwxm]

One thing that make the real world not pure mathematical is that even without autocorrelation, your performance (or ITM)varies over time. The losing stream is closely related to your ITM and its distribution. Therefore, you might improve your game or losing the edge over time, which affects your losing streams. /images/graemlins/cool.gif

[rachelwxm]

Just had my first 8 games losing stream and I now play 180 games so far with ITM 42%. Even I am not happy about my bankroll swing, I am happy that the numbers work out fine with my math. /images/graemlins/cool.gif