Upswings, Downswings, & the Normal Distribution

[shermn27]

Something I was thinking about the other day, if one plays solid theoretical poker, should this minimize the downswings and maximize the upswings?

In other words, are the biggest upswings larger than the biggester downswings.

Finally, another way of asking the question is, are the profits from each sessions (call every 100 hands a session) normally distributed?

For some reason I have mixed thoughts. It seems natural that they would be normally distributed, with a mean say of 05/100 BB and an SD of 10BB. But it also occurs to me that playing good poker may (should?) maximize the upswings and minimize downswings such that the distribution is no longer normal and the SD becomes rather irrelevent.

Maybe someone with lots and lots of hands can break them down in to 100 hand segments and plot them on a distribution. Unfortunately, I don't have very many so I can't do it. Or maybe someone can just explain to me why it is normally distributed and why playing well does not maximize upswings and minimize downswings. Thanks for you thoughts.

[AaronBrown]

If the hands are independent then the results of 100 hands will be pretty close to Normal. That's a mathematical result known as the Central Limit Theorem.

However, the hands may not be independent. For example, you lose a big pot in a showdown to a lucky player. As a good player, you should be able to exploit this, and get him to overbet when you have the stronger hand. On the other hand, if you win big early, other players might tighten up or leave. Factors like this can make the distribution non-symmetric and thin-tailed compared to Normal. If instead you tilt, you could get fat tails and negative skewness.

[Orpheus]

I hope I am not misreading your OP, but I think there's a fundamental misconception there.

Solid, even 'perfect', theoretical play (if such a thing exists) does not guarantee, or even suggest a profit. Your longeterm profits come ONLY from the errors of your opponents. Solid theoretical play and good reads can only help you minimize your own errors, so that you lose less to them than they lose to you.

This isn't a pedantic point. Imagine ten identical people, or artificial intelligences with a high EQ (ability to recognize and manipulate human emotions) playing a million hands of (near) perfect stratgy. Obviously, they can't all win. In fact, if they are equally good, and play in a real world setting, they'd all be guranteed to LOSE money to the rake or hourly charge. [Real-life poker is less than zero sum, thanks to the 'skim')

I understand that this may not seem relevant to your questions, but if the above is true, the statistical structure of your wins and losses will be heavily influenced by the statistical structure of THEIR errors.

In other words, though some very sharp and good 2+2'ers will sometimes argue otherwise, you can only make limited predictions about your winrate or bankroll fluctuations, based solely on your own play -- they are not entirely "your" stats. This is much more obvious in other games: Team X may have different scoring patterns against Teams Y and Z, because they use a passing game, a zone defense, smell bad, etc. The common patterns vary between conferences or leagues -- and sometimes styles become "fashionable", changing the mix in a league in just a few seasons

It'd be hard to deny that different "leaks" or suboptimal styles with the same -EV can cause different patterns of loss. Some might be low variance (small additonal wins and losses), or high variance and uniform (win big and lose big, roughly comparable numbers of hands); others might be high variance and sporadic (you win much bigger pots than you lose, but win much less often than you lose). Different leaks almost certainly have different statistical "shapes".

If you think of your overall winrate as the sum of all the leaks at all the tables you play, minus your own leaks at those tables (minus the expenses of playing), then it seems that the "shape" of (e.g.) your bankroll plot or winrate would epend on your competition, and I *don't* think that is entirely random. I believe, for example, that there is real variation between the mix of styles at various internet sites, or online vs casinos, or at various casinos.

These differences can be self-reinforcing (if there is a lot of Style X at a given site, people with Style Y might lose quickly, and either change their style or leave), encouraged by cardroom policies (e.g. bonus/reward structures) or influenced by outside factors (e.g. Vegas casinos favored by the locals, who are more likely regular players, probably have a somewhat different mix than those 'tourist' casinos). They may even be personal (affected by YOUR table selection)

If my assumptions are correct, then identical players won't have identical patterns of upswings and downswings, even in settings where they may have the same longterm winrate. This means there may not be a single monolithic answer to your original question.

Of course, it is possible that there are so many bad players everywhere, in a standard mix of patterns, that any variation in player population is minor. That doesn't jibe with my own subjective experiences, but I may be wrong.

[AaronBrown]

I agree with this in principle, but think it won't matter much over 100 hands. For example, suppose you play roulette. If you bet $6 on Red or Black, you win $6 18/38 of the time and lose $6 20/38 of the time. If you bet $1 on a single number, you win $36 1/38 of the time and lose $1 37/38 of the time. The 6:1 bet ratio makes the standard deviations about the same.

Although the shape of these distributions are quite different, over 100 spins the results both begin to look bell-shaped and pretty similar. 66% of the time with colors, and 62% of the time with numbers, you'll make or lose less than $60. 26% of the time with colors you'll lose more, 26% of the time with numbers. You'll make more than $60 8% of the time with colors, 12% with numbers.

As long as the bets are independent, the long-term distribution of their sum depends mainly on the expected value and standard deviation of individual bets, not their shape.

However, you also mentioned self-reinforcing tendencies. These can skew the long-term distribution away from Normal. And I agree that is more dependent on your opponents than you.