Thinking about the "Long Run"

[joel2006]

I did a search for this but couldn't find anything, so any links would be appreciated. My question is does anyone know of any serious research done on what constitues the long run in poker generally and HE specifically. Not best guesses or reasonable surmises, but actual research (either math based or simulation-based). Let me clarify, when I say the "Long Run" I'm not talking about the distribution of starting hands, but rather results (wins/losses of monies). Several simulations (done with Wilson Turbo Texas Hold'em) I have seen show that for a table of players with all the same skill level, results don't even out over a million hands. That is, when the results are due strictly to randomness, 1 million hands is not enough for wins/losses to converge, even though starting hands have evened out at this point. There is always one big winner and one big loser with the rest bunched in the middle. So clearly one guy is running good/bad over a million hands. Previously (it seems to me) when people talked about the 'long run' they often assumed that starting hand distribution and results distribution were the same or similar. But this is less true for HE than for the games which were popular then (5 Card Stud, 7 Card Stud). And the WTTHE results show this consistently. If we assume a B+M pro plays 35 Hands/Hour for 40 hours, for 50 weeks a year, that would be 35 hands X 2000 hours a year+ 70,000 hands a year. If he played for 40 years this would be 2,800,000 hands in his lifetime. Let's say a serious Rec Player plays half that for 1.4 million in his lifetime and a casual player plays around 1/3 of that for 935, 000 hands. Is there any actual evidence that any of them will reach the long run for 'Results"? Also, a related question is how many times must a player play a specific hand (say AA) for his results with that hand to converge? Or what is a large enough sample size to say that his results with that hand are reliable, is it 30,000, 50,000, or 100,000? Or is it even more? Can this question be answered by calculations alone? or must one run simulations? Any serious answers/attempts to tackle these questions are seriously appreciated.

[JackStrap]

learn about standard deviation or read malson book gambling theory and other topics its should help you answer this

[joel2006]

I underdtand STD DEV. I've read Mason's book and there is nothing in there about this. If I wasn't clear above, I'm looking for "actual research". things like this article.

[AaronBrown]

I think there's a subtle reason your question cannot be answered.

It's easy to say what happens if, say, everyone at a table of ten bets $1 on each hand and deals it to showdown. Each player's expected profit is, of course, zero, with a standard deviation of $3*SQRT(N), where N is the number of hands. The more you play, the more the lucky player's result will differ from the unlucky player. The lucky player's profit as a fraction of total amount bet will go to zero, but her total profit will only increase with more hands.

Now good cards is only a minor part of the luck in real poker. If you get terrible hands all night, you won't be the big loser. That will be the guy who gets very good second-best hands, and has everyone fold when he gets his good hands. Looking at it another way, my first calculation just showed the risk in the number of pots you win, there's a lot of additional risk from how big the pots you win and lose are.

So simulating with computer poker players, as Lou Kreiger did, will show more risk than just the card luck. He found about $10,000 separating lucky from unlucky players over 60,000 hands of $20/$40. That suggests the standard deviation was about $4,000 over 60,000 hands, or $16 per hand. We still expect the difference between the luckiest and least lucky player will increase with the square root of the number of hands played. With 16 times as many hands (960,000), we'd expect $40,000 to separate the big winner from the big loser.

The idea of the long run is that luck averages out relative to skill. So you need unequal players. If I figure to make a profit of $1 per $20/$40 hand, then over 60,000 I don't worry about a $4,000 standard deviation. My long run is more like 1,000 hands, because my expected return and standard deviation are about equal after 300 hands. I'm very unlikely to be a loser over 1,000 hands. But if my positive expectation is $0.01 per hand, my $600 expected profit over 60,000 hands looks pretty small compared to my $4,000 standard deviation. I need half a million hands for my expected profit to equal one standard deviation, so a million or two hands is the long run for me.

The trouble is, once you consider differences of ability, the long run becomes hard to define. The long run assumes each hand is independent, but good players adapt to the cards. If a good player is getting poor cards, he knows that other people will overestimate his strength, he can use this to his advantage. If he loses a big pot to a one-out on the river, he'll use that to his advantage as well. The question isn't whether a good player will win, but how long it will take her to get the bad player's money. So it's not standard deviation that matters, unless it's a tournament or some other situation with a fixed end-point.

[joel2006]

Aaron, thanks for the response. What about this question-in Lou's last simulation he used three solid players, three avg players, and three rocks. obv the solid players were all winners but at different rates, even though the differences were strictly due to chance. So if 60,000 hands isn't enough for the winning players results to converge, how many hands does it take? This question, (I think) brings us closer to an idea of how long the long run actually is.

[AaronBrown]

But my point is it depends on how much better the good players are. And it's even more complicated than that, because good players learn and adjust their play.

For playing one hand, or one hundred hands, mean and standard deviation matter. But when you ask about longer stretches of hands, the non-independence becomes important. The long-run is really a concept for independent events, like roulette or flipping coins. In poker, it's hard to define.