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#21
07-17-2005, 03:20 AM
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Re: A Less Obvious Martingale Fallacy
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It makes no difference whether bet sizes increase at random or due to long losing streaks. [/ QUOTE ] If Cosmic Casino has to repay the entire amount lost to them plus 1 unit when the gambler finally wins, then how are they making money here? 
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#22
07-17-2005, 03:21 AM
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Re: A Less Obvious Martingale Fallacy
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Doesn't your so-called "theoretical zero" correspond to a so-called "theoretically infinite dollar amount (loss)" on the other side of the coin? [/ QUOTE ] No, because I'm going to use my lifespan as a definition of infinity [img]/images/graemlins/wink.gif[/img] . For arguments sake, if infinity is a lifetime -- and you play 24 hours a day until death -- you can find a number of progressions with a finite bankroll that will give you any confidence level you want, and I do believe in a theoretical zero. While a long sequence of losses is required to lose, it isn't to win. I'm saying neither one will happen. [ QUOTE ] wouldn't there be a few who would bear a commensurate loss to your summed profits by hitting that nasty old extreme negative end of the tail? [/ QUOTE ] I'm saying no. That's why nobody will play this game. I'm not suggesting the house edge ever goes away, it's negative expectation -- but if the player can control the wagering to this degree, that expectation will never be realized.
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#23
07-17-2005, 03:23 AM
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Re: A Less Obvious Martingale Fallacy
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This is only possible because of the unlimited bankroll, but given that unlimited bankroll is a condition of this problem, I still don't see how our gambler can fail to bust the bank 1 unit at a time. [/ QUOTE ] You have to be careful to distinguish between (a) an extremely large but still finite bankroll, and (b) a literally infinite bankroll. For any extremely large but finite bankroll, we can do the math directly and see that we are still -EV (in roulette or any other game where each trial is -EV). If you jump to a literally infinite bankroll, then the casino also has to have a literally infinite bankroll (to cover our biggest possible wagers). So the answer to why you won't eventually bust the casino one unit at a time is that you can't bust an infinite bankroll one unit at a time. You can keep subtracting one from the casino's 'roll all you want, but you will never bust it.
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#24
07-17-2005, 03:27 AM
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Re: A Less Obvious Martingale Fallacy
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No, because I'm going to use my lifespan as a definition of infinity [/ QUOTE ] This doesn't really work, though. As an example, 5 years ago it would have been extremely rare for any professional poker player to play more than 125,000 hands in a year. It is not uncommon for us to be able to play 4, 5, or even 6 times that number of hands now. Does that mean that the lifetime of a pro poker player has been lengthened? Why put an artificial limit on this?
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#25
07-17-2005, 03:34 AM
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Re: A Less Obvious Martingale Fallacy
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If you jump to a literally infinite bankroll, then the casino also has to have a literally infinite bankroll (to cover our biggest possible wagers). So the answer to why you won't eventually bust the casino one unit at a time is that you can't bust an infinite bankroll one unit at a time. You can keep subtracting one from the casino's 'roll all you want, but you will never bust it. [/ QUOTE ] Fair enough, but if the game is actually +EV for the casino, then there should be a finite bankroll capable of defeating the unlimited Martingale bankroll, no? If not, then you need to conclude that the Martingale played with an unlimited bankroll actually does overcome the house edge, regardless of its size. I know this is true because you will eventually run into a series that overwhelms the finite side.
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#26
07-17-2005, 03:40 AM
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Re: A Less Obvious Martingale Fallacy
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[ QUOTE ] It makes no difference whether bet sizes increase at random or due to long losing streaks. [/ QUOTE ] If Cosmic Casino has to repay the entire amount lost to them plus 1 unit when the gambler finally wins, then how are they making money here? [/ QUOTE ] What you will see as you look over all the tables at the Cosmic Casino is a lot of players who are ahead a little bit, and a few players who are stuck huge amounts. The identities of the players who are ahead and who are stuck keep changing. But as time goes on it becomes more and more likely that the Casino will be ahead overall. What matters is who's rack the money is in at any point in time. Overall, the liklihood is that the Casino will have more money won and its racks than the players. As far as an individual player goes, if you're going to count yourself as being ahead after a win, you also have to count yourself as being behind when on a horrendous losing streak. You may spend more Time being ahead, but if you weight your time spent ahead or behind by the Amount you are ahead or behind, you will spend more Weighted Time being behind. PairTheBoard
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#27
07-17-2005, 03:40 AM
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Re: A Less Obvious Martingale Fallacy
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Fair enough, but if the game is actually +EV for the casino, then there should be a finite bankroll capable of defeating the unlimited Martingale bankroll, no? [/ QUOTE ] Yes, any finite bankroll will defeat the Martingale. A finite bankroll on the part of the casino means a finite betting limit -- so the player will be unable to keep doubling his bets past a certain point. [ QUOTE ] If not, then you need to conclude that the Martingale played with an unlimited bankroll actually does overcome the house edge, regardless of its size. [/ QUOTE ] With literally infinite bankrolls, neither party has an edge. If either side has a finite bankroll, then the house will have an edge (assuming a game like roulette where the house has an edge on each individual trial).
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#28
07-17-2005, 03:42 AM
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Re: A Less Obvious Martingale Fallacy
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Why put an artificial limit on this? [/ QUOTE ] Because there are limits. Using infinity is an artificial abandonment of constraint -- and why you can't define the expected outcome of the game. My calculation would require a definition of zero (impossibility), and I have seen people define it differently for different purposes. But once that limit is defined, we can apply it to a lifetime of play and achieve that confidence level. I don't know what to do with infinity [img]/images/graemlins/confused.gif[/img]
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#29
07-17-2005, 03:46 AM
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Re: A Less Obvious Martingale Fallacy
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Yes, any finite bankroll will defeat the Martingale. A finite bankroll on the part of the casino means a finite betting limit -- so the player will be unable to keep doubling his bets past a certain point. [/ QUOTE ] This is not a finite bankroll for the casino. It is a finite bankroll for the player. The assumption was that the player is allowed to wager any amount he likes and chooses to wager 1 unit after a win and double the previous wager after a loss. A finite bankroll for the casino would be a limited amount of capital that they could pay winning wagers with. (ie: an individual player wins)
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#30
07-17-2005, 03:49 AM
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Re: A Less Obvious Martingale Fallacy
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This is not a finite bankroll for the casino. It is a finite bankroll for the player. [/ QUOTE ] Same thing. The betting limit depends on the lesser of the two bankrolls. So if the bank's bankroll is finite, then effectively so is the player's. He can't double up his bets anymore once he gets to the point where the casino can't cover his bet. [ QUOTE ] The assumption was that the player is allowed to wager any amount he likes and chooses to wager 1 unit after a win and double the previous wager after a loss. [/ QUOTE ] This requires an infinite bankroll for the casino as well. How can the player wager an infinite sum if the casino only has $100?
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