A probability challenge

[tipperdog]

While debating the wisdom of a "hit and run" strategy with a poker buddy, I posed the following challenge. I'm confident that I know the results, but I lack the brainpower/expertise to prove it. I thought that some of the regular posters in this forum might enjoy the challenge, and I would be grateful to anyone who could show/teach me how to solve this type of problem.

Here's the problem:

Player A and Player B are two longtime poker buddies who have decided to turn pro together. Both are excellent players with many thousands of hours as winning players at 30/60 limit HE, the game they plan to play professionally. Over an extremely large sample, Player A has a win rate of 1BB/hr, after rake, tips, etc. Player B is a better player, with a win rate of 1.66BB/hr, after all expenses. Both Player A and Player B have tight-aggressive styles that yield a standard deviation of 10BB/hr.

Both player A and player B have bankrolls of 300/BB. Both player A and player B have a "weekly nut" of $1,500/wk. Both plan to withdraw $1,500 from their bankrolls each week for living expenses, regardless of their performance at the tables that week. Both player A and player B will play 5 days a week, 52 weeks a year. For the sake of argument, we will assume they never get sick or take vacations.

Player A plans to play approximately 5/hrs per day. Based on his win rate, he believes that will be sufficient to meet his financial goals. Player B has a different approach. He plans to play each day until he reaches a profit of $300, whether it takes 5 minutes or 24 hours. When he reaches his goal, he quits for the day. We will assume (somewhat unrealistically), that player B has the capacity to play for 24 hours straight if needed to make his goal and that the quality of his play will remain constant for all 24 hours.

So here's the question: What is the likelihood that each player will go broke?

Also, if Player B increased his daily profit target to a certain number and left the excess in his bankroll, is it possible for him to have a lesser chance of going broke than Player A. What would that profit target be?

Both players' chances of going broke are 1 since they're both operating on a fixed amount of money, all that it takes is a sufficiently long bad streak (which is unlikely, but possible).

If the players are building their bankrolls while they play, their expected bankroll can grow sufficiently quickly to outpace the likelyhood of a sufficient bad streak so that they might have a lower expectation of ruin, but I don't particularly care to calculate where that occurs.

[Doc7]

When you say Player B's quality willl remain constant for 24 hours, do you also mean that Monday's quality of play is the same as Tuesday's? Or does he have days where he may be playing at a significantly lower or higher level (for 24 hours straight)

Because if that was the case, doesn't this fall under the "stay at the table as long as it is profitable and leave when it isn't" ? If his quality of play has variance to it i would think that Player B's method is a much worse plan than Player A.

no mathematical proof.

[tipperdog]

[ QUOTE ]
When you say Player B's quality willl remain constant for 24 hours, do you also mean that Monday's quality of play is the same as Tuesday's? Or does he have days where he may be playing at a significantly lower or higher level (for 24 hours straight)


[/ QUOTE ]

I meant that Player B's quality of play would not deteriorate INTRA-session, but it also follows that his quality of play remains constant INTER-session. It's somewhat unrealistic, but let's assume that player B will be perfectly fresh and clear-thinking after 23 hours of play.

We'll further postulate that Player B begins each session at 12:01 a.m. If he plays for 24 hours straight (and thus, it's now the next day), he will start a new session immediately--again with no loss of play quality. We assume his win rate remains 1.6BB/hr with SDEV 10BB/hr regardless of how long he's been at the tables.

[Doc7]

Well then playing for 5 hands a day or 24 hours a day won't make a difference will it? He has the same likelihood of going broke no matter if he plays until he is up $300 for the day or if he plays exactly 1 hand a day, the only thing that changes is how long it takes him to reach $0.

not that I can tell you what his chances are of going broke, or what they are compared to player A. but i think that given what you just said, if he plays exactly the same poker all the time, his chances of going broke do not deviate no matter what his "plan" is, be it to play a certain number of hours or make a certain number of dollars per day.

at least that's what my intuition tells me.

edit:
So what I am saying is if Player A had the same win-rate and StDev as Player B (all factors are exactly the same) , but if player A played 5 hrs/day and player B played till he made 300/day or until new 24 hour period started, they both have the same chance of going broke, but I think player B would go broke sooner (playing more hands in the same amount of 24-hour periods, probably)

[AaronBrown]

A is simple. His chance of being broke after N hours is 2*Normsdist(-300/(10*SQRT(N))). His median survival time is 1,980 hours of play. Since he plays 25 hours per week, that's about a year and a half.

We can ignore the possibility of B going broke from his 300 BB bankroll in one day. The chance of that is less than one in a billion.

The question is what happens if B finishes up 24 hours without making $300? If he just keeps playing until he's up $300 times number of days played, then he is very unlikely to ever go bankrupt. But if he absorbs the shortfall from his bankroll and goes back to his rule of stop at $300 each day, he will eventually go broke.

He will have a shortfall about once every 33 playing days, and lose about 39 BB on the average losing day (that's counting the shortfall from $300, since he spends $300). That means he'll last about a year with average luck, less than A.

[tipperdog]

[ QUOTE ]
A is simple. His chance of being broke after N hours is 2*Normsdist(-300/(10*SQRT(N))). His median survival time is 1,980 hours of play. Since he plays 25 hours per week, that's about a year and a half.

We can ignore the possibility of B going broke from his 300 BB bankroll in one day. The chance of that is less than one in a billion.

The question is what happens if B finishes up 24 hours without making $300? If he just keeps playing until he's up $300 times number of days played, then he is very unlikely to ever go bankrupt. But if he absorbs the shortfall from his bankroll and goes back to his rule of stop at $300 each day, he will eventually go broke.

He will have a shortfall about once every 33 playing days, and lose about 39 BB on the average losing day (that's counting the shortfall from $300, since he spends $300). That means he'll last about a year with average luck, less than A.

[/ QUOTE ]

Thanks Aaron,

I'm actually somewhat surprised the results were that close. I would have thought player A would last FAR longer than his buddy.

Player A and Player B are really playing fundamentally different games.

Player A is playing "one long game." His results will fluctuate--bankroll goes up, bankroll goes down--riding the wave of variance until an unusually large wave (bad luck) wipes him out.

Player B is playing a series of "one day" games. Because his daily withdrawal rate is $300, the same as his daily goal, he can AT BEST break even on a given day. He will do this most days. But his bankroll can only move down over time. There's no "up and down" for Player B...just an unsteady descent.

My point (which you've helped me prove, thanks), is that the better player (Player B) will easily have the worse outcome if he chooses such a foolish approach to managing his bankroll.