Variance, winrates, and expectation (long)

[beachbum]

We talk a lot about downswings in this forum, mainly because SH play has a bigger variance inherent to it compared to full ring play. After I hit my first 300 BB downswing last month, I almost felt relieved. My thoughts were, "well, it's going to happen eventually so might as well learn to deal with it." It was almost like I had popped my poker cherry, so to speak.

So that got me thinking on how normal this magnitude of a downswing is. We all talk about winrates, not being results-oriented, sample sizes, etc. alot. I thought it might be interesting to graph some different winrate and variance scenarios to better visualize these concepts. Check out the following graphs:



The 2 graphs are basically the same. The only difference is that the first graph illustrates variance considering a 1 BB/100 winrate, and the second graph shows 2 BB/100. The axes are labeled showing big bets earned and # of hands played. Both graphs are built on the same scale so it's easy to compare the two. Also, I used a standard deviation of 15 BB/100 for both.

The black line in the center shows a consistent zero-variance winrate (for reference). The colored lines on both sides of the winrate line show 2 different sets of limits concerning variance. Basically, your results should fall within the outer bands 99% of the time, and fall within the inner bands 95% of the time.

From these graphs some interesting observations can be drawn.


Downswing Severity

Looking at these graphs, the assumption is that the player is playing at a certain "skill level" (whether at a 1 BB/100 or 2 BB/100). Of course, this skill level will never be constant through a sample number of hands. It will fluctuate based on the typical variables of table selection, seat selection, whether the player is tired, distracted, on tilt, etc.

Take a look at the 2 BB/100 graph. If a player played at a solid 2 BB skill level for a sample size of let's say 20,000 hands, he really shouldn't have a very big downswing. If the player was able to maintain this quality of play even through lots of bad beats and cold cards, only about 5% of the time should he experience a downswing of greater than 110 BB's.

However, if this player's play now degrades to a 1 BB skill level, his downswings can be much more brutal. The same 5% of the time his downswing can get past 200 BB's. But a smaller percent of the time the downswing can get to be pretty high even surpassing the magic 300 BB number.


Break-even streaks

I think the break-even streches really get more magnified as a player's play degrades. In the 2 BB graph, it's fairly unlikely that a player will have a breakeven streak of more than 20,000 hands. However, if he's only playing at a 1 BB level, this same percentage of the time this breakeven streak can surpass 80,000 hands. Hell, it could be possible to be down after 100,000 hands and still be an overall 1 BB player.

Seeing the difference in these 2 graphs makes me want to focus even more on improving my skill level. Even though a 1 BB winrate is solid, there will definitely be times that will shake your confidence, to say the least. This tends to be a snowball effect too as bad results tend to bring about bad play and further degrade a player's winrate to even a negative number if tilt becomes a problem.

I guess bottom line we should continue to focus on our quality of play instead of getting so caught up with winrates and money won. It'll make life a lot easier [img]/images/graemlins/smile.gif[/img].


Predictability of Income

I found it interesting to see that as the number of hands increases, the range of money won continues to increase. Now this is different from expected winrate. Obviously, we know what our winrate is with more accuracy the more hands we play.

But look for example at the 1 BB graph. After 100,000 hands, we know that we'll have made somewhere between 0 and 2000 BB's about 95% of the time. This is a pretty big range. For those of us that play poker for a source of income, this unpredictability can be a bit trying at times.

Consider a full time pro who plays 20,000 hands per month of 10/20 at a skill level of 1 BB. At the end of the year, 95% of the time his earn will be between $19,200 and $76,800. Although it's only a small percent chance, it will be hard to feel good about your year if you've only made low $20's and felt you've played decent poker. FWIW, change this winrate to 2 BB and this lower bound jumps to $67,200.


I just thought some of these observations were pretty interesting and wanted to share. Please correct me anywhere I'm wrong or made a mistake. If any of this has helped you become less results-oriented and more focused on improving, then it makes the time spent putting this post together worth it. Just don't sit at my tables [img]/images/graemlins/wink.gif[/img]


Chris

[TheMetetron]

tl;dr


does that work in this forum?

[RunDownHouse]

[ QUOTE ]
Take a look at the 2 BB/100 graph. If a player played at a solid 2 BB skill level for a sample size of let's say 20,000 hands, he really shouldn't have a very big downswing.

[/ QUOTE ]
How do you explain all of the empirical evidence to the contrary? Tilt? How does a player running ultra-hot for 1k hands, then ultracold for 3k hands fit into this?

[Surfbullet]

[ QUOTE ]
[ QUOTE ]
Take a look at the 2 BB/100 graph. If a player played at a solid 2 BB skill level for a sample size of let's say 20,000 hands, he really shouldn't have a very big downswing.

[/ QUOTE ]
How do you explain all of the empirical evidence to the contrary? Tilt? How does a player running ultra-hot for 1k hands, then ultracold for 3k hands fit into this?

[/ QUOTE ]

This graph is a 95% + 99% confidence interval. Even the 2BB/100 winner could have a 40k breakeven stretch, according to the 99% CI. Short-term variance is much more severe than this pretty graph, as well.

Furthermore, a SD of 15 was used for this approximation. Most SH players on this forum experince a SD of between 16.5-19 depending on playstyle. This opens up the possibilities of running good/running bad to a far wider magnitude.

Surf

[Surfbullet]

Hey Beachbum,

Excellent post. I'd love it if you could run another set using a SD of 17 and a SD of 19, since a SD of 15 would be more indicative of, say, a 5/10 full ring player.

If you were to run those other 2 sets we should have them stickied at the top of the forum, with a notification "If you're going to make a downswing post, READ THIS FIRST."

Great job.

Sur

[krishanleong]

[ QUOTE ]
Great job.

[/ QUOTE ]

I agree. I'd also like to see the graphs for 17 and 19 STDV.

Krishan

[joseki]

I think you're underestimating the 1%.

Let's say you play a +2BB/100 game but still drop 100BB in your first 10k hands. Looking forward to your next 10k hands, you face a real possibility of losing another 100BB. Same with the next 10k.

Looking forward those events are very unlikely to occur back to back, but having it happen the first time does not reduce the probibility of it happening again (obviously).

All I'm saying is that those runner-runner two-outers do come in, despite them falling outside the 99% confidence interval.

[ QUOTE ]
I think you're underestimating the 1%.

Let's say you play a +2BB/100 game but still drop 100BB in your first 10k hands. Looking forward to your next 10k hands, you face a real possibility of losing another 100BB. Same with the next 10k.

Looking forward those events are very unlikely to occur back to back, but having it happen the first time does not reduce the probibility of it happening again (obviously).

All I'm saying is that those runner-runner two-outers do come in, despite them falling outside the 99% confidence interval.

[/ QUOTE ]

I think you're over-estimating the 1%. OP isn't looking ahead 10k hands at a time. You are giving a scenario that fits your argument (lose for 10k then lose for another 10k then lose for another 10k). Independently, these aren't near 1%, but cumulatively they are. Everytime you look at something AFTER it has already happened, you are cutting the odds of the whole outcome down by that factor.

I'm not the best with explaining math, so I hope that made some sense.

[joseki]

Also, dude, these graphs address the probibilty of future outcomes starting "now". Every time you're delt cards that's one "now". From the begining of each hand, if you know that your winrate will remain constant and you know how much variance to expect, then you can plot these curves to get a probabilistic view into the future.

From a different perspective, when you look back on 100k hands there are 100k obervations and you can expect 1k of them to fall outside the 99% CI.

Am I wrong?

[ QUOTE ]
Also, dude, these graphs address the probibilty of future outcomes starting "now".

[/ QUOTE ]

This is the only way to calculate future outcomes.