gambler's ruin

[bztips]

Almost as confusing as the logic that something with >0 probability may not happen during an infinite set of trials.

In probability, never say never, and never say always.

Probability theory RARELY says anything interesting about prob=0 or prob=1. For the present discussion, the interesting question is how likely it is that bankruptcy may occur. Of course the answer depends on the EV of your bets (and their variance) relative to your bankroll.

If you're making lots of -EV bets that are a significant fraction of your bankroll, then yes you should probably worry about the potential for going bankrupt. But if you're making +EV bets, then the probability of going bankrupt is smaller (again depending on their size and how much +EV they are). Is there a chance you will still go bankrupt? Of course, but that doesn't mean that you will go bankrupt with probability=1 due to "gambler's ruin", which is what the poster was arguing.

[Optional]

Fair enough.

[Bsims]

I guess I should start buying lottery tickets because "eventually" it has to hit.

Actually if you could participate in the lottery an infinite number of times, you would indeed hit the lottery. In fact you would hit it an infinite amount of times. Of course you would lose an infinite amount of money because your cost would vastly exceed your winnings.

[Flight]

Sample size is the primary source of confusion in this debate.

To assume that the house/book has infinite bankroll is not a good assumption. It is very much finite and we see it manifest itself in player limits. Higher limits are extended to players with -EV betting patterns. I have not seen higher limits extended to players making +EV wagers, in fact quite the opposite. +EV bets are viewed by sportsbooks as acts of war on their business.

One last point: Comparing finite elements to non-finite elements leads to errors in math, thinking, and judgment.

[Jmoiz888]

Just read through this

[tomcowley]

Here are some questions.

If "there is a possibility" that an event will occur, does that mean the probability of that event occurring is >0?

If the probability of something happening is >0 and you have an infinite number of trials, then will the event occur?

If the event will occur, then can you say that the probability that the event occurs is 1?

You're trying way too hard- the general point doesn't involve "almost surely". If you start with $100, and play an infinite series of coinflips, losing costs you $1, winning wins you $1000, what are the odds of ever dropping below $100 on that infinite sequence? The odds, obviously, are "just over 50%", and not anywhere near 100%. If you lose the first flip (50%), you drop below, but if you win the first flip, then you're a gigantic underdog to ever drop back to $100.

[bztips]

You're trying way too hard- the general point doesn't involve "almost surely". If you start with $100, and play an infinite series of coinflips, losing costs you $1, winning wins you $1000, what are the odds of ever dropping below $100 on that infinite sequence? The odds, obviously, are "just over 50%", and not anywhere near 100%. If you lose the first flip (50%), you drop below, but if you win the first flip, then you're a gigantic underdog to ever drop back to $100.

Thanks, this is what I was trying to say; your example makes it very clear.

[Flight]

You're trying way too hard- the general point doesn't involve "almost surely". If you start with $100, and play an infinite series of coinflips, losing costs you $1, winning wins you $1000, what are the odds of ever dropping below $100 on that infinite sequence? The odds, obviously, are "just over 50%", and not anywhere near 100%. If you lose the first flip (50%), you drop below, but if you win the first flip, then you're a gigantic underdog to ever drop back to $100.

Actually, the geometric series

1/2 + 1/4 + 1/8 + ...

sums to 1


I see that you are trying to make a point by playing a +EV game where you are offered 1000/1 odds, but that is confusing. In a 50/50 game, if you play it "infinite" times, the probability of "going negative" is 1.

[tomcowley]

Actually, the geometric series

1/2 + 1/4 + 1/8 + ...

sums to 1


I see that you are trying to make a point by playing a +EV game where you are offered 1000/1 odds, but that is confusing. In a 50/50 game, if you play it "infinite" times, the probability of "going negative" is 1.

It shouldn't be confusing- it's the whole point of advantage gambling. Make +EV bets and win in the long run. Your 0EV game is one of the reasons that the optimal stake for a 0EV position is 0.

[jcubs55]

Haven't read through the proof yet, but curious what causes a gambler to go bust in a scenario where his EV = 0? My gut says something about poor bankroll management, but even that gives him as good a chance to over-bet himself out of a losing streak as it does bury him. Is it that the gambler will bet until he has nothing left to gamble?

Variance. If the house has an infinite bankroll, variance can't bankrupt them, regardless of how far variance sways against them. But against the player with the finite bankroll, it is only a matter of time before variance gets to a point where the player loses his finite amount of funds. If he were to also have an infinite bankroll, this point of bankruptcy will now be irrelevant because he still has funds to continue gambling at 0 EV and over time the law of averages will overcome variance and head back towards a point where neither the house or player are up/down money (back to the breakeven point of origin).

[squallsquall]

At this rate you would need to lose 50 consecutive wagers to go broke................but the beauty of Kelly is that over time when your bankroll shrinks so do you wager sizes so really you would have to lose hundreds of wagers in a row in order to wipe out your bankroll when betting with a perceived 2% edge.

If your bet size is always a given fraction of your bankroll, you will never go broke even if each bet is a 100% guaranteed loss. Still, I'm having a hard time seeing the magic in that method.

As I understand, Kelly's strengths lie at optimizing growth rate, not at making it impossible to go broke.

[sharpcat]

If your bet size is always a given fraction of your bankroll, you will never go broke even if each bet is a 100% guaranteed loss. Still, I'm having a hard time seeing the magic in that method.

As I understand, Kelly's strengths lie at optimizing growth rate, not at making it impossible to go broke.

You obviously do not understand Kelly.

If a bet is a 100% guaranteed loss than Kelly is going to recommend that your bet size be 0% of your bank roll, if your bet has a 50% probability of winning and the line you are being offered is +100 than Kelly is going to recommend that you bet 0% of your bankroll because you have no edge.

Kelly minimizes your bankrolls decline the same way that it maximizes its growth.

[bztips]

Kelly minimizes your bankrolls decline the same way that it maximizes its growth.

Not exactly. You can certainly find other strategies that have lower variance than Kelly (for example, fractional Kelly); they lower the prob. of a big decline (and, obviously, also lower the likelihood of big gains).

[That Foreign Guy]

Not betting at all minimises potental for bankroll decline (unless you bring expected growth in to play) but assuming you're going to bet, you should bet more when the event is more likely and you have a bigger edge (Yay Kelly)

[JayTrotter]

I missed something.

We are betting in an environment or marketplace, where the sports book evaluates players and risk. This has nothing to do with gambler's ruin. When we wager, we are not playing against the house. This is similar to horse race wagering. Losers pay the winners, and winners pay the vigorish. The game however is fixed. The house wants to "level" the playing field so that no one wins, because if there are no winners, no one gets paid, except for the house. They also like it when there is more money on the losing side of the wager, for the simple reason that they want to keep more of the money being bet..

Gambler's ruin is based on an assumption that is not true in the case of sports betting. The casino is not specifically competing with you. unless, they have evaluated that they need to. Even so, A sports book operation has an objective to make money through influencing the betting patterns and volume of its' consumers. They are not in the business of breaking individuals, unless those individuals are worth breaking. The sports book wants to make ROI and manage risk.. period. The reason they limit winning bettors is because they can. The system is set up so that most will lose, their tactics just insure that the casinos can make a lot of money, and fully manage risks.

[JayTrotter]

flat out maximize return. minimizing risks is like clipping coupons. a waste of a Sunday.