is there a good reason to limit the spread at a table?
I'm wondering about this rule in particular:
* The system must have a betting range of no more than 1 to 1028 units
Why make that stipulation? Is there some way of beating those games if a wider spread is offered? I personally don't think so, but am interested.
I run a casino that offers a factor of about 18 billiion in the spread between min and max bets, and am wondering if that makes it exploitable. Should I limit the spread to 1028x like in the contest to be safer? Actually I also allow players to bet 0 (to win 0), so in that case the spread is infinite...
It seems to me that a negative expectation game should be unbeatable no matter what range of stakes is offered.
Thoughts?
Quote: dooglusWizardOfOdds used to run a "Betting System Challenge" in which he offered $20,000 to anyone who could demonstrate a winning betting system on craps, roulette, or baccarat. The rules are listed here:
https://wizardofodds.com/gambling/betting-systems/challenge/
I'm wondering about this rule in particular:
* The system must have a betting range of no more than 1 to 1028 units
Why make that stipulation? Is there some way of beating those games if a wider spread is offered? I personally don't think so, but am interested.
I run a casino that offers a factor of about 18 billiion in the spread between min and max bets, and am wondering if that makes it exploitable. Should I limit the spread to 1028x like in the contest to be safer? Actually I also allow players to bet 0 (to win 0), so in that case the spread is infinite...
It seems to me that a negative expectation game should be unbeatable no matter what range of stakes is offered.
Thoughts?
A larger spread doesn't make the game beatable, but it does allow you to raise the variance of your results, which in turn requires a larger number of simulations before there is a high probability that your results are within a certain range of the expectation.
In other words, the point of this is to show that, in the long term, your betting system will lose. The larger your spread, the further away the "long term" can be.
Quote: GWAEplus I think a martingale system might win if you have unlimited units to bet.
This isn't really true.
The (mathematically correct) way to state this is that if you have unlimited units to bet, and you use a martingale system, your expectation is undefined. The problem is that you are trying to evaluate in infinite sum that doesn't converge. Not all infinite sums are defined (and this one isn't). There are plenty of mathematical "paradoxes" that are based on this concept.
Quote: AxiomOfChoiceA larger spread doesn't make the game beatable, but it does allow you to raise the variance of your results, which in turn requires a larger number of simulations before there is a high probability that your results are within a certain range of the expectation.
In other words, the point of this is to show that, in the long term, your betting system will lose. The larger your spread, the further away the "long term" can be.
That makes sense.
Since the challenge was to provide a system which won after a billion bets he needed to limit the spread to prevent a system from winning by getting lucky.
My site has a 1% house edge on all bets, has processed almost 420 million bets, and has held just 0.23% of the total amount wagered. That's almost certainly a result of the massive spread. The vast majority of the bets have been very small, meaning the relatively few large bets define the profitability.
Increasing the minimum bet probably wouldn't have had much effect on the absolute profit, but would have reduced the total number of bets, making the 0.23% profit look more reasonable.
Quote: AxiomOfChoiceCan I ask what your site is? How do you allow bet spreads in the billions? A billion (American billion) cents is still 10 million dollars. Are you allowing people to wager for 1/1000 of a cent or something?
It's a Bitcoin gambling site. Bitcoin is divisible to 8 decimal places, so I allow a minimum bet of 0.00000001 BTC (or 0, but the payout is also 0 when you win).
Individual bitcoins are worth roughly $800, or 80k cents. So the minimum bet is worth roughly 1/1250 of a cent, pretty much exactly what you guessed. Weird.
There's no fixed maximum bet; I set the maximum profit per bet to be 0.5% of the site's total bankroll ("half Kelly"), and so currently the most you can win per bet is 180 bitcoin, or 18 billion times the minimum bet.
It's at Just-Dice.com if you want to take a look. Just please don't use any profitable betting systems on it. ;)
Quote: onenickelmiracleThis reminds me of the gambit from revolutionary days where the costs were so high a price to play wouldn't be low enough. It has to do with doubling up from $1. What's that called again?
I can't understand what you're saying, but it made these two concepts come to mind:
Quote: onenickelmiracleThe rice problem is very similar. What I was thinking of had to do with doubling up from a dollar allowed possibly to infinity, but the cost to play was so high the fee to play was just too much and nobody could afford to Pay out winnings without transferring the casino over. Just about the same idea as the rice story. I think it's something like only royalty could afford to play.
Might you be talking about this http://en.wikipedia.org/wiki/St._Petersburg_paradox . Basically the expected value of the bet is infinite but the amount people would be willing to pay to play is relatively low.
The martingale on a 1 to 18b spread is obviously still negative expectation for the player but you might need several lifetimes for the house to win that last big bet to actually show a profit on it.
Yeah that's it. Thanks.Quote: TwirdmanMight you be talking about this http://en.wikipedia.org/wiki/St._Petersburg_paradox . Basically the expected value of the bet is infinite but the amount people would be willing to pay to play is relatively low.
Quote: dooglushttps://en.wikipedia.org/wiki/Wheat_and_chessboard_problem
I like to pose the question as, start with one mill [0.10 penny. ] and the choices are
*by the 64th square typical person would have the money to place there
*only Bill Gates would have the money
*the world would have the GDP to place the money there
*not even the world would have the money
Quote: dooglusIt's a Bitcoin gambling site. Bitcoin is divisible to 8 decimal places, so I allow a minimum bet of 0.00000001 BTC (or 0, but the payout is also 0 when you win).
Individual bitcoins are worth roughly $800, or 80k cents. So the minimum bet is worth roughly 1/1250 of a cent, pretty much exactly what you guessed. Weird.
There's no fixed maximum bet; I set the maximum profit per bet to be 0.5% of the site's total bankroll ("half Kelly"), and so currently the most you can win per bet is 180 bitcoin, or 18 billion times the minimum bet.
It's at Just-Dice.com if you want to take a look. Just please don't use any profitable betting systems on it. ;)
I have to say, this is pretty cool. I particularly like the provably fair part.
I took a look at the stats and I couldn't figure out why your profits are so much lower than expectation. I was expecting to see something like a bunch of people martingale-ing you, or something similar (maybe that is happening but it's not clear from the stats). Of course in the long run this wouldn't matter, but with such a large bet spread the long run can be a very, very long time.
I assume you have access to all the bets ever made. Can you see how many standard deviations below expectation you are? (ie, add together the variances of every bet ever made and take the square root)
Quote: AxiomOfChoiceI have to say, this is pretty cool. I particularly like the provably fair part.
I think the coolest part is that the bankroll is dynamically crowd-sourced.
Any player can, at any time, lend any part of their bankroll to the site to use as its bankroll. They then get a proportional share of the profit or loss of every bet placed by all the players on the site. I take 10% of any new net profits they make at the end of each week.
In this way I'm able to run the site without risking my own funds. At one point the bankroll was over 60k BTC, and Bitcoins were trading for over $1000 each at the time. So the site's bankroll was worth over $60M, all contributed anonymous by people who don't know me.
"Provably fair" is pretty much a standard requirement of all Bitcoin gambling sites. It's so easy to do that if a site isn't doing it, it's considered they must have something to hide. And since the majority of Bitcoin users tend to be maths/tech geeks, enough of them understand what provable fairness is about that it's really useful to have it. Even the ones who don't understand it get a positive feeling that it must be fair or the ones who do understand it would have started shouting about it by now.
Quote: AxiomOfChoiceI took a look at the stats and I couldn't figure out why your profits are so much lower than expectation. I was expecting to see something like a bunch of people martingale-ing you, or something similar (maybe that is happening but it's not clear from the stats). Of course in the long run this wouldn't matter, but with such a large bet spread the long run can be a very, very long time.
I assume you have access to all the bets ever made. Can you see how many standard deviations below expectation you are? (ie, add together the variances of every bet ever made and take the square root)
How do I calculate the variance of a bet? If someone bets 2 units with a 16% chance of winning and wins, what is the variance of that bet? And if they lose?
Quote: dooglusHow do I calculate the variance of a bet? If someone bets 2 units with a 16% chance of winning and wins, what is the variance of that bet? And if they lose?
Whether they win or lose, the variance is the same.
http://en.wikipedia.org/wiki/Variance
If X is a random variable representing the possible results, then var(X) = E(X^2) - [E(X)]^2 (where E() is expectation).
So, what that means is, take the expected value of the square of the result, and subtract the square of the expectation.
Your example: Bet = 2 units. P(winning) = 16%. If they win they get 10.375 units (right? that would be a 1% house edge). So:
E(X^2) = (.84 * (-2)^2) + (.16 * 10.375^2) = 20.5825
[E(X)]^2 = (-.02)^2 = .0004
Var = 20.5821
S.D. = 4.5367...
Variance is in units squared, which is kind of a useless measure. It is useful only because it is additive. So if 3 bets have variances of 5, 11, and 20, the variance for all 3 combined is 36 (and so the standard deviation of all 3 bets combined is 6)
Note that Var(aX) = (a^2)Var(X), so you can figure out the variance as a function of the chance of a win, and then multiply by the square of the bet amount (note that we will take the square root of the variance when determining the standard deviation, so the standard deviation will scale linearly with the bet amount, and we will be back to units rather than units squared)
So...
Compute the variance of all bets.
Add them all up
Take the square root of the total
That is your cumulative standard deviation. So you can figure out how many standard deviations away from expectation your results are.
Now, I'm not a statistician, and this is where my skills are failing me. I suspect that this isn't actually that useful of a measure, since your possible outcomes may not follow a normal distribution. I think that they have to tend towards a normal distribution if you have enough bets (I never really understood the central limit theorem -- this would be a great time for one of the stats people to jump in) but I think it's possible to structure the bets against you in a way that you are very likely to be slightly below expectation (with the tradeoff being that you have a small chance of being way above expectation -- this is basically how martingale works)
Quote: dooglusI think the coolest part is that the bankroll is dynamically crowd-sourced.
Any player can, at any time, lend any part of their bankroll to the site to use as its bankroll. They then get a proportional share of the profit or loss of every bet placed by all the players on the site. I take 10% of any new net profits they make at the end of each week.
What if they lose money during a particular week?
Quote: AxiomOfChoiceWhat if they lose money during a particular week?
I remember how much profit they have paid commission on, and don't charge it twice.
If they earn 10 BTC in the first week, I charge 1 BTC.
If they then lose 5 BTC in the 2nd week, I charge nothing - they take the whole 5 BTC loss.
If they then win 6 BTC in the 3rd week, I consider the first 5 BTC of that 6 to be recovering previous losses, and only charge 10% of the new 1 BTC of profit.
Back in September (I think) we had a huge whale of a player come to the site and spam thousands of max bets for over 12 hours. He bet over a million BTC in those 12 hours, and somehow managed to quit while ahead, leaving the site's profits very negative.
Quite a few investors (that's what we call the people bankrolling the site) are still recovering the losses they made on that day, and so they're still not paying me any commission whether they have winning or losing weeks. I still get to charge some investors commission; mostly the new ones who weren't around in September, but also some who were lucky enough to time getting in and out of the bankroll on that crazy day that they actually made a profit 'day trading' the whale by being invested for more of his losses than his wins.
tl;dr if they lose, they take the whole loss; if they win, I take 10% of their profit, if it really is new profit
