A question for you math geeks....

[Daddys_Visa]

In the NVG thread there is a long discussion about whether poker is actually gambling. There are several subdiscussions in the thread, one of which I would like answered by you theoretical math types. I made the following statement which has been challenged, and while I intuitively believe it to be correct I cannot make a very articulate proof of it. I stated:

Every poker will eventually bust out if the following conditions are met:

1.) Every player plays for an infinite amount of time.
2.) There is an infinite amount of money available to be won in the poker community.

I believe this to be true regardless of the players edge in the game (assuming they are not 100% to win each hand), their bankroll, or the stakes they play.

Is this true?

[DrVanNostrin]

It's not true. A similiar question is if you flip a fair coin and take two steps forward every time it comes up heads and one step back every time it comes up tails, are you a lock to end up at least 5 steps back at some point?

I can't find the thread but the answer is no. It's a lot like the idea of convergence. That is, it's possible to sum an infinite series of numbers and get a finite number.

[Daddys_Visa]

[ QUOTE ]
It's not true. A similiar question is if you flip a fair coin and take two steps forward every time it comes up heads and one step back every time it comes up tails, are you a lock to end up at least 5 steps back at some point?

I can't find the thread but the answer is no. It's a lot like the idea of convergence. That is, it's possible to sum an infinite series of numbers and get a finite number.

[/ QUOTE ]

I like your example. Very simple, yet I still have the same conceptual problem. What I don't understand is if with each flip you have a finite chance of entering a streak of tails long enough eventually get you to 5 steps back, how can infinity not ensure that will inevitabley happen? I can see that once you get several thousands steps forward, the chances of ever getting to 5 steps back become infinitesmally small, but finite nonetheless.

[plexiq]

No, its not always true.

It depends a lot on the players bankroll management. If the player has an edge and does not move up, he *always* has a chance to go broke of less than 100%, even over infinite time.

The thing that is probably confusing you is:
Just because a player has a non-zero chance to go broke in the future at any point, does not mean he is 100% to go broke over infinite time. This is just like summing 1/4+1/8+1/16+...+1/2048+...+1/inf converges to 1/2. Despite summing up an infinite number of non-zero terms, the sum will never reach the 1/2-bound.) I think thats the root of your confusion about risk of ruin.

I tried to be as non-formal as possible, hope this clears things up for you.

Edit: The 2 Posts above me came up while i was writing, and DrVanNostrin already explains it well. But i keep my post here anyway, i guess.

[tarheeljks]

why do you think this makes sense intuitively. (not a troll)

[Daddys_Visa]

[ QUOTE ]
No, its not always true.

It depends a lot on the players bankroll management. If the player has an edge and does not move up, he *always* has a chance to go broke of less than 100%, even over infinite time.

The thing that is probably confusing you is:
Just because a player has a non-zero chance to go broke in the future at any point, does not mean he is 100% to go broke over infinite time. This is just like summing 1/4+1/8+1/16+...+1/2048+...+1/inf converges to 1/2. Despite summing up an infinite number of non-zero terms, the sum will never reach the 1/2-bound.) I think thats the root of your confusion about risk of ruin.

I tried to be as non-formal as possible, hope this clears things up for you.

Edit: The 2 Posts above me came up while i was writing, and DrVanNostrin already explains it well. But i keep my post here anyway, i guess.

[/ QUOTE ]

No, you didn't put too much math in your response. I have enough college math under my belt to understand what you are saying and it makes more sense now. Ya, the whole convergence thing is reminding me of all those derivative and integration classes I tried to forget. [img]/images/graemlins/wink.gif[/img] thx

[Daddys_Visa]

[ QUOTE ]
why do you think this makes sense intuitively. (not a troll)

[/ QUOTE ]

The way I was thinking about it is that if on each poker hand your risk of ruin is a finite number no matter how much you are up or what your edge is or how many hands you play, an infinite amount of hands played would eventually result in a losing streak bad enough to break you. I mean, when you talk about infinity, even a 1,000,000 consecutive losing hand streak is still very possible.

I seem to recall Sklansky posting a question to the forum like 5 years ago. He gave a scenario in which a casino offered an even money game (say flipping a coin). Assuming the casino had an infinite bamkroll would they make a profit at such a game? The answer was yes, but I forgot how it was proven.

[DarkMagus]

I can't remember how the proof goes but I remember from a physics class that a 1-dimensional random walk will eventually visit all points on the number line.

From wikipedia: (http://en.wikipedia.org/wiki/Random_walk)

[ QUOTE ]
Suppose we draw a line some distance from the origin of the walk. How many times will the random walk cross the line? The following, perhaps surprising, theorem is the answer: for any random walk in one dimension, every point in the domain will almost surely be crossed an infinite number of times. [In two (or perhaps three) dimensions, this is equivalent to the statement that any line (or plane) will be crossed an infinite number of times.] This problem has many names: the level-crossing problem, the recurrence problem or the gambler's ruin problem. The source of the last name is as follows: if you are a gambler with a finite amount of money playing a fair game against a bank with an infinite amount of money, you will surely lose. The amount of money you have will perform a random walk, and it will almost surely, at some time, reach 0—and the game will be over.

[/ QUOTE ]

I think this still applies even if the player moves down in limits when he loses a significant chunk of his roll, because there must be some minimum betting limit.

Now, whether the same thing applies if the player has an edge (as in a good player at a poker game), I'm not sure. Intuitively, I think it would, but I can't prove it.

[AaronBrown]

Your memory is correct, a random walk with constant standard deviation and no trend visits every number. So if you keep playing at constant standard deviation and no edge, you eventually go broke.

However, even a small positive edge changes that. You still have a chance of going broke, but it's not 100%. Even without an edge, you can play forever if you reduce standard deviation as you lose.

The classic "gambler's ruin" result is that however big your edge, you eventually go broke if you raise stakes as you win, but don't cut them as you lose. This is how most gamblers act. As they win, they raise the stakes to take advantage of their hot streak. When things turn against them, they don't reduce stakes, because they want to make the loss back.

A lot of poker players are really gamblers at heart. They move up in stakes and take more chances when they are winning, but when they start to lose they don't want to go down in stake, and they can't bring themselves to play more carefully. However good a poker player you are, you will go broke if you act like a gambler.

[EasilyFound]

[ QUOTE ]

Every poker will eventually bust out if the following conditions are met:

1.) Every player plays for an infinite amount of time.
2.) There is an infinite amount of money available to be won in the poker community.


[/ QUOTE ]

Feeling a bit ornery today, and someone has to dumb down this thread. Since neither condition can be met, isn't this a pointless question?