Monte Carlo Methods or Why It's a Bad Idea to Go to the Casino
easylang.online
easylang.online
There's two main effects:
- Negative expected value (EV)[1]. The games are set up in a way that on average you're likely to lose money. This is true for every game where you play against the house, otherwise the game wouldn't be there. Poker is different because your EV depends on how you play relative to other people.
- Bankroll management and risk of ruin[2]. Even in a fair game, if you start with a short bankroll, you're likely to go broke. For example, let's say two of us flip a completely fair coin and bet $1 on each flip. The expected value for both players is $0. Now if one of us starts with $5 and the other $1000 and we play repeatedly, the person with the smaller amount is almost guaranteed to go broke in a long series of flips. This what happens when you play against the house and it's the main reason why people lose money when they gamble in casinos.
For number 2 it depends on strategy. The strategy you outlined has you risk $5 to win $1000. Of course in a fair game you will go broke most of the time. For every time you win you should lose 200 times to come out even. But you can easily take the reverse strategy. Bet $5 if you win, leave. If you lose double your bet, if you go down $1000 dollars leave. In this case you are almost guaranteed to win. But the times you do lose you lose enough to make up for all of your wins.
EV is EV, it doesn't change based on who has the larger bankroll. The casino doesn't magically get an edge because they have more money. They get the edge because the game odds are in their favor.
The key distinction is whether you count the EV of a single wager, or the expected growth of compound returns.
As you say, the single-wager EV is independent of bankroll provided you can afford it in the first place. This is also what most people mean when they say EV, and indeed the common mathematical definition of it.
However, skilled risk takers know that the arithmetic EV isn't what matters more generally. What matters in the long run is geometric EV, or expected growth of compound returns.
And for geometric EV, bankroll absolutely matters. The larger your bankroll, the better your geometric EV. I suspect this is what the parent comment referred to.
(However, when the wager shrinks in comparison to your total wealth, the arithmetic EV approaches the geometric EV by Taylor series approximation. For "everyday affairs" (as I think Bernoulli put it) you can think of them as equal.)
A casino would much rather play against a player with $1,000,000,000 than a player with $5. Sure the player with $5 is going to go broke nearly 100%. And the player with more money will never go broke betting $5 at a time, they will die from old age way before that happens.
The fact that a player with only $5 has almost a 100% chance of ruin is absolutely not why a casino is making money. A casino is always going to want a player with a lower risk of ruin than a higher one. When a player goes broke they can no longer make money from them.
Do people really think a casino would prefer someone going broke 100% of the time with $5 then a player going broke 0% of the time with $1,000,000,000? That they are making money because of players losing their bankrolls and unable to bet?
The way I understand it, they're not saying "casinos make money because their counterpart is going broke" (and you're right that they don't.) They're saying "in the casino vs. player situation, the player will almost always run out of bankroll before the casino does, and therefore the casino does not risk ruin, even in highly volatile games."
There's the players making small bets, they win a bit, lose a bit, and at the end of the day they may go home with a bit more or a bit less money than they came with. The casino wins on average because of the 0.
Then there are the gamblers. They come to the table, make a few large bets, and within a few minutes they lost their monthly salary (or whatever was the limit on the ATM). They lost because the player will continue playing until they can't.
I'm pretty sure that you could make a completely fair roulette, and the house would still win big every day. I'm not sure why there aren't any fair roulettes, though.
It's a random walk, and it's much more likely to take you to -$5 way before it takes you to +$1000.
Actually the largest Powerball at $1.586 billion was split between 3 people [1]
[0] https://chance.amstat.org/2020/02/lottery-ticket/jackpots-ev... [1] https://apnews.com/article/archive-6d98eff765c9e2f4b68367131...
There’s some ethical concerns where the gaming industry (and liquor industry) will market more towards problem users or in ways to generate more problem users, but buying $20/lotto tickets a month isn’t a guaranteed start of a problem, nor is having wine alone sometimes.
Yes. Also briefly addressed in the post.
The simulation already comes close to the calculated value.
print 10 * 10000 * (18 / 37 - 19 / 37)
> Bankroll management and risk of ruin
No. The loss results purely from the EV (in roulette -0.027) . And that is my total bet * -0.027. No matter if Bezos is playing or I am.
> The loss results purely from the EV
This is true for a single spin, it is not true for a sequence where once you run out of money, you stop playing.
> > The loss results purely from the EV
> This is true for a single spin, it is not true for a sequence where once you run out of money, you stop playing.
If you have to stop gambling because you lost your million dollars, you didn't lose because you don't have any more money. You couldn't turn it around with another 100 million either (in a non-hypothetical casino).
Corollary: the best way to win is the one where others don't care or don't even realise you have won. Try to find a niche others don't care about and find in it regular, small wins. Don't go into a highly contested arena and try to win big. Even if you think you have won, you'll never get paid that way.
I've been scammed at foreign casinos before in the Caribbean, but never in the US.
Again if the casino can reject your win for any reason after declaring you won then they can take back the money for any reason period. Therefore there is no reason to ever play at such an establishment.
And the player CAN challenge a loss. They too can file a complaint with the commission, who will investigate with video footage and any other evidence, and by mission is designed to be on the player's side if the evidence for not paying out isn't solid.
They aren't going to willy nilly void payouts because of the scrutiny it would bring.
A more typical spread of 10-to-1 ekes out an expected value of ~0.5% (see https://wizardofodds.com/games/blackjack/card-counting/high-...). When I was counting, I could barely hit $15/hr expected value. Hardly worth the effort, I found it very demanding to keep track of the count at Vegas dealer speeds and play perfect blackjack for hours on end.
...and your bankroll...
> will allow.
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You can still go bust on a 1.5 % edge, after all, if you overbet.
This sounds counterintuitive to many so it's worth considering for a bit.
The law of large numbers says that the number of successes divided by the number of tries converges. It doesn't mean the number of successes converges to expectation - quite the contrary.
For example when you flip a coin and go up a unit every time it's heads and down one unit every time it's tails your expected distance to 0 is sqrt(n) where n is the number of flips. Similarly when you play poker your expected distance from real EV to what you actually get is going to increase the more you play.
With this mind it's not correct to say the amount is already close to calculated results with more tries as the article says. Try it, you will get numbers farer away from the expectation the more times you play!
If you're playing a perfect good blackjack strategy, even without counting cards, you have a small edge over the house (if I remember my Thorpe correctly.) Let's call it 0.5 %.
So if you play 100 hands, you can expect to win 51 of them. However, sometimes you might win 59, or 63, sometimes only 45 or 37. But you'll be reasonably close to 51. If you won fewer than 36 hands or more than 66 I'd be surprised. So in the worst case you win 15 hands fewer than your expectation, and in the best case you win 15 hands more.
If you instead sat day and night and played 10,000 hands, you'll still be expected to win the same fraction: 5,050.
But you might win 5,073, or 5,168. Or win 5,012, or win 4,931. With this large number of hands, you might well win 150 hands fewer than you expected to, or 150 hands more!
Relatively speaking, you're closer to the expectation with the additional trials. With 100 hands, you might win anywhere between 36 % and 66 % of them. With 10,000 hands, you'll probably win between 49 % and 52 % of them.
But when you're playing 100 hands, the difference between 36 and 51 % is only 20 hands. When you're playing 10,000 hands, the difference between 49 % and 51 % is 200 hands! In terms of actual absolute money won or lost, you're more likely to end up farther away.
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And the reason, of course, is that while the expectation grows linearly when you increase the number of hands, the variation shrinks with the square root of the number of hands.
The reason they converge in the law of large numbers is not that the variation shrinks as you play more hands -- the reason they converge is that the expectation outgrows the variation, which makes the latter seem small relative to the former.
This is also why the distance traveled in a random walk is dominated by variation in the short term and bias in the long term. When you have a small enough time step, the square root is greater than the line.
A coworker told our team about a puzzle relating to sizes of pizzas. Most people came up with clever solutions that all depended on things like uniform density, perfect circularity, etc. Probably good approximations to real world pizzas. But one of us suggested a Monte Carlo based method and the beautiful thing about it was that it would work even without all those assumptions. As long as the pizzas have an area when viewed from above, it works.
I don't think this is accounting for the role that risk plays into the addiction.
That’s a special case of “big advantage in skill” in the sense that the WSOP/TV boom drew a bunch of players with way more aspiration than skill/experience and the games were easily beatable by players of typical home game winner level of skill as a result.
But you can still get cheap gross rooms at Circus Circus!
For a professional dealer, good chips, and a continuous flow of players, that seems like a pretty fair price to pay.
Or just paying for entertainment. Craps pass line is under 1.5% per bet placed with even money odds if you’re backing up your bet. For a $15 pass line bet you’re paying the house $0.25 per bet. Even at 10 bets/hour it’s $2.50/hour for entertainment, which is super cheap. And toss in a couple free drinks and it comes out to be very cheap entertainment.
Edit: the real key way to view it is that much like bartenders, casino staff are fundamentally providing recreation. The ones that are good at their jobs will provide a solid experience for a reasonable price.
On the other hand anyone with a gambling addiction needs to stay away. A good rule of thumb is to treat your entire trip bankroll as entertainment budget that you’re willing to spend all of.
[1] For all I know they might not even care if the counter’s base bet is melons, but my experience is limited to black chip range since losing won’t hurt and it’s enough to get the dopamine hit plus sign off on nice comps.
If you bet $10 on black and win straight away, earning $10, and go home with $20, and stay home for the rest of your life and never walk into a casino again, then yes, you won. But you're not really a gambler. You're actually the opposite of a gambler, one that never enters a casino again. Of course you gambled $10, but the amount is meaningless.
If instead you gambled a meaningful amount, say $50k, then indeed it'd be truly gambling something meaningful. But if you'd lose, you'd have to double it to $100k. If you lose that, $200k. It spirals out of control quickly if you're actually betting meaningful amounts.
And the amounts must be meaningful. After all, if you bet $10, lose a few and eventually win and go home with $80, the strategy requires you to never gamble again. If you come back because $80 isn't meaningful, you're really just starting over at a meaningful amount, that could make you win enough to never make you come back. But at that point, you're talking about significant risks (e.g. betting $50k and losing 4 coin unfavourable (47% / 53%) tosses in a row, and you've lost a total of $750k and need to risk losing another $800k to walk away with a profit of just $50k, by flipping another unfavourable coin toss)
If you were Jeff Bezos, of course you could bet a million and walk away with an eventual win before running out of money. But the amount of money isn't meaningful to Bezos. In the same way a kid with $40 of savings could bet $50 cents and walk away with 50 cents of profit by winning the 5th round. But that doesn't make any meaningful difference if it ends there. If the kid comes back to gamble, the strategy falls apart, as the house always wins in the end if people keep coming back.
And that's precisely why the strategy doesn't actually work for anyone. To win any meaningful amount, you'd have to bet a meaningful amount. And to do that, is incredibly risky as there's a good chance you'll run into your limit before you win. And even if you win, that experience will likely pull you back in to try to repeat it, the gambler mindset eventually will lose.
How quickly do you run into your own limit, and the limit of the casino? After all, the odds are against you.
Technical note: your final game winnings are never more than the first game winnings. That's why the first bet must be meaningful.
Say you bet $10 and lose, bet $20 and win, you made $10 profit. Say you lost the $20 too and bet $40, you again still made $10 as you made 80 - (10+20+40). Suppose you lost the $40 too, and the $80, and the $160, and the $320, and the $640, and now bet $1280 and win $2560... subtract 1280+640+320+160+80+40+20+10, and again, you only walked away with $10.
In other words, to walk away with something meaningful like $10k, you may end up having to wager 1.2 million after 7 coin toss losses.
Are there really casinos with no maximum bet? Usually there is a maximum so you run into the problem that at some point you can't double anymore and have to take the loss, even though you would have enough money left.
Also beyond roulette I've heard of various martingale variations in different gambling games. For example, a common system here in Greece is play martingale on draws on a soccer league (like the World cup or the Euro Cup): Pick the first game and bet on draw (no matter what the teams are). If you win you'll get 3x (that's what the draw usually gives); if you lose, double it and bet draw at the next game. This is a good system if you can afford doubling. I remember a Euro some years ago where there were like 15 games without a draw ; consider that if you started with 1 euro you'd need to bet ~ 32 0000 euros in the 15nth game ...
and over all I feel like it's no different from going big on 1 bet, you just save time.
After a few hours of botting there was always streak of 10, 11, 12, 13 and more "bad" choices in a row
One guy was found with a crowbar in his in his head in front of his house. It didn’t go well for the others. I think they are all died middle age.
In other words you have to be pretty rich already and have a deep bankroll to get the breakeven comps.
Otherwise, consider your losses the fee for the entertainment. :)