Small Stack vs. Big Stack

[Mathemagician]

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This is what I suspected you meant and it is just ridiculous. If I have a $10,000 bankroll in a game with $2 and $4 blinds in which I am a reasonable favorite over you, then my risk of ruin is certainly negligible, no matter what your stack size is.

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If his stack size is infinite your ruin is guaranteed. Also, as the ratio ofhis stack size to yours increases towards infinity, your risk of ruin increases towards certainty.

M

[jason1990]

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If his stack size is infinite your ruin is guaranteed.

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I'm not going to play an infinitely long session with the guy. I'm not going to play until one of us is broke. I'm just going to play my normal game for a while and try to win some money off him (after all, I assumed I am a favorite over him). I imagine, as usual, I would stop when I got tired or had something better to do with my time. If I'm properly bankrolled for an "ordinary" game, wouldn't I also be properly bankrolled to play against him? Should I really be worried about going broke in this situation? Should "hugging my nuts on the rail" really be a concern for me here? I don't see it.

[pzhon]

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[ QUOTE ]
This is what I suspected you meant and it is just ridiculous. If I have a $10,000 bankroll in a game with $2 and $4 blinds [and $200 maximum buy-in] in which I am a reasonable favorite over you, then my risk of ruin is certainly negligible, no matter what your stack size is.

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If his stack size is infinite your ruin is guaranteed.

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While that is technically true, it is not true in a useful sense. It is true if, having doubled up so many times that you have millions of dollars on the table and only a few thousand dollars left in your pocket, you are compelled to try to double up the money on the table again as a slight favorite. It is more reasonable to assume that you are following the Kelly Criterion or something similar, and will quit when the amount of money you have on the table is a significant fraction of your bankroll.

Suppose you have the ability to get all-in as a 3:1 favorite. The Kelly Criterion (maximizing the expected logarithm of your bankroll) suggests that the optimal amount of your bankroll to wager is 1/2, and that you should walk away if asked to wager more than about 84% of your bankroll. You only need to double up 8 times to have $51,200 on the table, which will represent enough of your bankroll that you should quit.

The probability of doubling up 8 times is about 1/10. The probability that you double up 8 times in a row in your first 25 attempts (without cutting your $10k in half) is about 92.8%, at which point you have $56,200-$61,000 and quit.

The 7.2% of the time your bankroll gets cut in half, you should lower your sights, and should quit after you have $25,600 on the table, doubling up only 7 times. You succeed about 1/7.5 of the time, and have 13 attepts before your bankroll is cut in half again. You end up with $28,000-$30,400 84.5% of the 7.2%. You fail again only 1.1% of the time. At that point, you should continue, but suppose you quit.

By using the Kelly Criterion, which many people find too aggressive, your bankroll is cut to $2400 only 1.1% of the time. 98.9% of the time, you end up with at least $28,000, and your average ending bankroll is over $57,000. This is a great deal.

The amount your opponent has over $100k is irrelevant because you will sensibly quit when you have 5/6 of your bankroll on the table.

[jason1990]

Thanks for this informative post. You mentioned some people find the Kelly Criterion too aggressive. Are there any commonly used alternatives?

[pzhon]

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Thanks for this informative post. You mentioned some people find the Kelly Criterion too aggressive. Are there any commonly used alternatives?

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To be consistent, people can maximize their expected utility, where utility is some concave function of the bankroll (negative second derivative). Of course, people are not always consistent, but that's another issue.

Some people use a fractional Kelly system. (See this paper for some analysis.) If the Kelly Criterion says to bet x, they bet x/k for a fixed k. The linked paper mentions that the probability that you ever drop to a*original bankroll is a^(2k-1).

Instead of maximizing E(log(bankroll)), this corresponds to maximizing E(-1/bankroll^(k-1)). For k=2, it is half as valuable to double up as to avoid losing half of your bankroll.

[soah]

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[ QUOTE ]
This is what I suspected you meant and it is just ridiculous. If I have a $10,000 bankroll in a game with $2 and $4 blinds in which I am a reasonable favorite over you, then my risk of ruin is certainly negligible, no matter what your stack size is.

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If his stack size is infinite your ruin is guaranteed. Also, as the ratio ofhis stack size to yours increases towards infinity, your risk of ruin increases towards certainty.

M

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I have a five-figure bankroll to play 2/5 NL. There are hundreds of 2/5 NL games available, each of which is played at various times by hundreds of different players. Each of those players has thousands of dollars they can afford to lose at poker. Their combined bankroll is somewhere in the tens or hundreds of millions of dollars. I can play 2/5 NL every day for the rest of my life and still never run out of opponents. This does not mean that my ruin is a virtual certainty.

There are two reasons for this. The first is because I am only buying in for a small portion of my bankroll. Once a day, I quit and go home, and the next day I buy in again for just a small portion of my bankroll. So I might briefly have $2000 on the table, but at the end of the day, $1500 goes into the bank and the next day I start over with $500 on the table. I have a significant skill advantage over my opponents so my bankroll is growing continuously. Each day my risk of ruin gets smaller. How much money my opponents have is irrelevent. It does not matter at all to me if I play against the same guy every day who has $10 million on the table, or if I play against thousands of different opponenents who each have $1500 on the table. I'm never putting my entire bankroll into play. If it were true that a player was certain to go broke if their bankroll was drastically smaller than their opponents', then there would be no such thing as a professional poker player, because all poker players would be broke. This is obviously not the case.

The second reason is because I am not playing infinitely long. If I have a million dollars, and I flip a coin for one dollar, then I would eventually go broke, if I played infinitely long. However, if I only flipped the coin 999,999 times, then my risk of ruin is exactly zero. As a poker player, I am dealing with a finite amount of playing. I will play until I don't want to play anymore... not until I've bankrupted every single opponent I can find, or lost my own roll in the process.