Question for the Wizard: Gambler's Ruin

Looks like we are to "ask the Wizard" here now.

Your Wisdom, sir, continues to amaze. I have a one that has been bugging me, and I apologize in advance if you have answered it before, but I have used the search function at your site and can't find that you have. The "probability of Ruin" in the context of a particular situation is addressed there many times, but not "Gambler's Ruin" in the bigger concept that one of the ways a Casino always wins is that it has the bigger bank. This does play out all the time when a gambler has set a limit that is near to the "up and down" oscillation that happens; even if it were possible to place an even bet, the Casino would frequently bust the gambler, whereas for all practical purposes there is no way that the Casino could ever suffer such a thing.

We also see this played out in TV poker in no limit tournaments. For example, as I remember, Jamie Gold in the 2006 World Series of Poker had amassed such a lead that he started to tell the last opponents he couldn't lose. All he had to do was wait for a hand that had a reasonable chance of winning, then force someone who had called him to go "all in". He realized he could survive these "all ins" but of course it was Ruin for the opponent whenever Gold won such a hand. In a poker tournament elimination is everything.

Now, I can no longer find this, but I saw it said somewhere that in fact Gambler's Ruin is *not* a means for the Casino to make more money since somehow it all evens out. But I believe in fact Casinos actually keep statistics on the "hold" they have on various players they give big comps to, and it seems to me "hold" is closely related to Gambler's Ruin.

Gambler's Ruin is sometimes presented by certain elements of Society as a warning to the public. Don't gamble, you can't win, this is another reason.

So my questions to you are, do you buy into the concept that it isn't just the odds that are in favor of the house, but also the size of the house Bank? Do you say the "warning to the public" is invalid? Isn't it true that one side vulnerable and the other not to Ruin just has an undeniable effect? Is it possible that this cannot be proved mathematically but as a practical matter it nonetheless does play out?

That last is a hell of a thing to ask the Wizard! Thanks in advance to your thoughts on it.

The way this question is usually put is whether or not the casino would win with a game with zero house edge, or even a player advantage. The asker will incorrectly assume that because the casino has the larger bankroll, the player will eventually go bust. I disagree. If a game had zero house edge, and the player’s bankroll was p, and the casino’s bankroll was c, and they played until one went bust, the chances of the player winning (casino going bust first) would be p/(p+c). The expected profit of both sides would be zero. I could prove this with collapsing sums, but forgive me if I skip the math. Problem 116 of my mathproblems.info site shows how to do it. I claim this would also apply to two or more equally skilled poker players, absent collusion.

The expected profit of the casino is simply the product of the house edge and total amount bet. If the player runs out of money, then that is a bad thing for the casino, because the house edge is applied to a less total bets. The casino would prefer the player keep digging himself deeper in the hole.

Quote: Wizard

The way this question is usually put is whether or not the casino would win with a game with zero house edge, or even a player advantage. The asker will incorrectly assume that because the casino has the larger bankroll, the player will eventually go bust. I disagree. If a game had zero house edge, and the player’s bankroll was p, and the casino’s bankroll was c, and they played until one went bust, the chances of the player winning (casino going bust first) would be p/(p+c). The expected profit of both sides would be zero. I could prove this with collapsing sums, but forgive me if I skip the math. Problem 116 of my mathproblems.info site shows how to do it. I claim this would also apply to two or more equally skilled poker players, absent collusion.

The expected profit of the casino is simply the product of the house edge and total amount bet. If the player runs out of money, then that is a bad thing for the casino, because the house edge is applied to a less total bets. The casino would prefer the player keep digging himself deeper in the hole.

I must bow to your superior knowledge, of course; I guess I can tell from your answer this is a long settled issue. I have to tell you there was something convincing about watching Jamie Gold seem to pull it off as an advantage. Possibly when elimination is the goal, it can be a factor, but of course he first had to build up that huge bank. Thanks for replying.

Just to jump in here and maybe state the obvious...what Gold did was purely psychological in the spirit of the game. The math would not support his claims, but it was a great way to push the table talk to lead players to think they were desperate.

in the '09 WSOP, I think Ceda had around 1m in chips on the final table and almost busted out several times...but came back to win the entire thing.

-K