Monte Carlo methods – Why it's a bad idea to go to the casino

I have a pet peeve about Monte Carlo methods; although it might be more fairly characterised as a rookie mistake I saw once.

MCM are not strong if the tail variance isn't an important feature of what is being modeled. I've seen simulations where the modeler starts with an analytic model - from which they could trivially calculate the mean and variance of a KPI - then used a MCM simulation to find out essentially what the mean and variance of the KPI.

So I get that this code is just for illustrative, educational purposes. That is fine. Well done Christof, thanks for the contribution to education. But if anyone is actually using MCM to simulate well known probability distributions they should really put the effort in to learning how to work with well known probability distributions. I feel pretty confident that simulating the Bernoulli in a professional setting is a mistake, because I've seen it done and it was a mistake. For amateurs who aren't confident with math then they can use MCM if they like; but statistics is dangerous and they should be aware that they are using the wrong tool for the job and it is a tell they will get other things wrong. You get a lot of insight from a good analytic model.

If there is some interesting tail issue (eg, maybe after 3 heads there is guaranteed to be a tails and your insurance contract pays out on a tails) then sure, use MCM. Great tool if you have the right problem.

If you want to really know why you shouldn't gamble (for profit... gambling for fun, with discipline, can be satisfying), read this quintessential book:

Scarne’s Complete Guide to Gambling

What you will probably take away from this book is that the house wins when you win. And the reason they win is that they don't pay actual odds.

Take the roulette wheel as a simple example. There are 37 (or 38 in the case of 0 and 00 wheels) slots. If you bet on a single number, they pay 36:1.

TFA mentioned doubling down when you lose. That's known as the Martingale Strategy. As banks and real estate lenders discovered in 2007, regardless of your model, there will be some condition where the near-impossible will happen.

If you want to use the roulette wheel again as an example, and even if you want to ignore the green 0/00 slots, you have half black and half red. You can simply play one color and double your bet if you lose... until you finally win it all back.

The problem there is that while it's true that for millions of spins, about half should land black and half should land red... but in short periods it is possible for there to be 10 or 15 or 20 or more spins of the same color. We can all do exponential growth, so it's easy to see how doubling down will empty your wallet eventually.

But just to be sure that strategy won't work, most casinos will set a table limit. That limit will be designed to prevent you from doubling your bet too many times (5-7ish times? depends on the place). Of course, if you discuss in advance with the casino, you may be able to set a personal table limit based on your initial bet. So you can bet 1 million on the first spin in order to set your table limit to 1 million. But that's truly a gamble.

There is one element you can control when gambling: that is when you increase or decrease your bets, and ultimately when you stop playing.

Yes, you can count cards in blackjack; but that's generally frowned upon (where "frowned" means you will be escorted out and told to never return -- and they are very serious). Even levels of indirection of counting such as employed by the fascinating experience of the MIT crews of yore (https://en.wikipedia.org/wiki/MIT_Blackjack_Team) have become obvious. So the short of it is, unless you only play a little, bet big, and get lucky for your limited period (and then stop so you don't get banned), you will otherwise lose more than you win.

Now for my personal opinion :). Casinos used to be a fun place. Now they are so optimized and commercial that they are like Walmart. Yes you can go, but you'd probably rather not. For modern thrills, we now have crypto exchanges with 50:1 or more leverage.

I was able to play martingale in a casino in a video game, so a limited wallet wasn't my problem, and I made money in the long run, but it exposed a subtler problem with Martingale: You only ever win your initial bet back; the rest is covering losses. If you have the cash to withstand 20 losses, then the initial bet, in comparison to your stake, is basically a rounding error. Who cares about winning $1 when you have $1 billion to martingale it?

Yes, that is also a problem. When you lose, your risk increases exponentially. Given an unlimited wallet and no casino rules, that's ok. But the real world unfortunately doesn't work like that. (And if you had an unlimited wallet, who cares if you lose?)

No matter what you bet, your take is only equal to your initial bet * number of rounds/hands played.

Really dependents on your philosophy. Would you rather do error propagation using this method http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm

or would you rather perturbate your variables and resolve mean and variance. My philosophy drives me towards the computational approach over the analytical mathematical approach.