Effect of player busts on casino profits from machines

If a casino were to offer a guaranteed 100% payback slot machine, over the long run both it and the players should break even. However, due to variance, some players will lose their bankrolls and walk away losers. Does this increase the casinos' profit, or will they break even in the long run because, regardless of who is playing the machine or how much down time it has, it's really all one long session? Thanks.

The latter.

Against one player, risk of ruin is a big factor in a 100% payback game. Against many players, the combined bankroll means no risk of ruin for the group. Players as a group should break even.

What you're referring to is called Gambler's Ruin. Basically, in an even game, the person with the largest bankroll will eventually win. You're correct in that it's one really long session and therefore this would not be a profitable game for the casino even with the Gambler's Ruin aspect. To test that I ran a sim of 10,000 players playing a coin flip game for $1 until they'd played 10,000 games or had lost $100 (Ended up being 85 million total flips). Below are the results; the Win/Loss is within expected variance:

Total Win/Loss: -808
Players that Lost their $100 Bankroll: 3117
Coin Flip Wins: 42468895
Coin Flip Losses: 42469703

I may not have thought this through thoroughly, but it seems that the standard deviation for 10,000 even-chance wagers for $1 by a player would be $1*sqr(10000)=$100. That would lead to an expected 15.87% probability of losing $100 (or more, if they weren't shut out). Thus, I would have expected only 1,587 of the 10,000 players to have gone bust. Do you have any thoughts on why so many more went bust in your simulation? Did I do the calculation properly?

Note: to those who saw the original version of this post, yeah I screwed that one up.

Quote: PopCan

Basically, in an even game, the person with the largest bankroll will eventually win. You're correct in that it's one really long session and therefore this would not be a profitable game for the casino even with the Gambler's Ruin aspect.

Only a player with an infinite bankroll. Largest doesn't cut it. Or you could never recover with 200,000 chips vs an opponent with 1,800,000 chips.
That's not the reason however.

The reason is that casino's winnings/player are capped at $100, while each player's winnings are not capped.

Quote: P90

Only a player with an infinite bankroll. Largest doesn't cut it. Or you could never recover with 200,000 chips vs an opponent with 1,800,000 chips.
That's not the reason however.

The reason is that casino's winnings/player are capped at $100, while each player's winnings are not capped.

This....

Think of the distribution as a bell curve, except cap the losses at $100. This would actually be worse for the casinos if the variance allowed for a decent percentage of winnings over the $100 cap, and would thus skew the curve...

Quote: Doc

I may not have thought this through thoroughly, but it seems that the standard deviation for 10,000 even-chance wagers for $1 by a player would be $1*sqr(10000)=$100. That would lead to an expected 15.87% probability of losing $100 (or more, if they weren't shut out). Thus, I would have expected only 1,587 of the 10,000 players to have gone bust. Do you have any thoughts on why so many more went bust in your simulation? Did I do the calculation properly?

Note: to those who saw the original version of this post, yeah I screwed that one up.

I believe the reason is that the standard deviation says 15.87% will END the session at $100 loss or more. During the session many of the people (apparently around 30% according to my sim) will swing below $100 at some point.

Quote: P90

Only a player with an infinite bankroll. Largest doesn't cut it. Or you could never recover with 200,000 chips vs an opponent with 1,800,000 chips.
That's not the reason however.

The reason is that casino's winnings/player are capped at $100, while each player's winnings are not capped.

Good point. I hesitated saying "will eventually win". I should have something more along the lines of "will most likely win".

Quote: PopCan

I believe the reason is that the standard deviation says 15.87% will END the session at $100 loss or more. During the session many of the people (apparently around 30% according to my sim) will swing below $100 at some point.

I suppose the shortcut analysis would be that ~half of the players who go bust would have done better after that (recovered a bit) if they had continued to play and ~half would have lost even more. Starting with your simulation results, that shortcut leads to an estimated 3,117/2 = ~1,559 who would be expected to be below -$100 after 10,000 wagers. That's reasonably close to the 1,587 that I suggested in my earlier post. Guess it makes a bit more sense to me now.

I'm not sure right now how I would go about using standard statistical distribution analysis (not a simulation) to estimate what percentage of players would bust out after fewer than 10,000 wagers. It's been too long since I have attempted more complex mathematics than what appears in my checkbook register.