In order to support the general belief that gambling is not a good idea, I have taken the trouble to run a few simulations to show you the development of traditional gambler’s careers. I have written a small programme that simulates the careers of three roulette players. These three roulette players play according to a certain system, each has a different one.
This is actually unrealistic. Because most players play rather chaotically. And the biggest problem with chaotic gambling can be that one loses track of the game and bets against oneself or makes bets and forgets them later, i.e. possibly does not collect the winnings.
But that is not the main problem. The main problem is that these players start to increase their stakes. And the higher the stakes, the higher the negative equity, i.e. the expectation of loss. Here I would ask you never to forget that it is possible to play on boiler errors or by observation, as you can read elsewhere. But the ordinary player does neither and has the predetermined disadvantage.
So the ordinary player raises the stakes and on top of that many people later play more than a single number, even 10 to 20, all on plein. At that moment, of course, the disadvantage becomes much greater. Because: if you play red or black, manque or impair, you only have a 1.35% disadvantage. On top of that, you also play 18 numbers. If you play numbers, you have already doubled the disadvantage to 2.7%. On top of that, you are almost obliged by the casino to give a slice to the employees if you hit a bull’s-eye (i.e. a plein, a number you have bet) (by the way, the employees get their wages exclusively from these tips, contrary to popular belief). This increases the disadvantage to a whopping 5.4%. And if you then actually already play 20 numbers with plein, then you have a stake that you could also play on a colour, but with 4 times the disadvantage.
In the long run, or even in the very short run, this will show up in your (God forbid), i.e. the player’s, wallet.
In my simulation here, the players, very disciplined, follow a certain strategy. I describe this strategy here:
The first player always bets one unit on Manque (if you like, he has 1000 euros and plays 10 euros of it per game). The second player always plays a number, each game with one unit, just like player 1, i.e. with 10 euros, he also has 1000 euros. The third player has heard about the doubling strategy. He uses it. He has 1000 euros, bets 10 of them on impair, 20 if he loses, 40 if he loses again and so on.
The diagram shows the result of a series of maximum 10000 attempts. You can follow the development of the players, at the same time look at their expectation curve. Please look:
The diagram shows the development curves of the three players and also their expected values in the development. The expectation values of player 1 and player 2 are moving very consistently downwards. Player3’s is a bit jagged as he varies his stakes (unlike the other two).
The player in purple is the doubling player. As long as he has money left, he wins a unit on every game he wins at all. So he has to go up in a jagged but steady way (unless he loses everything). The first deep cut comes when the first time his chance had gone for a long time. The second time, the end comes. His chance doesn’t come even in the 11th attempt (I looked in the results: the tenth time the zero came, he still had 301 before, after that 150.5, he bet that completely, then pair came again).
It’s a bit boring to explain the doubling strategy again, but I’ll do it anyway: You bet one unit. If one loses 2. If one then loses 4 units. One would get back 8. One then bet 1+2+4 = 7 and gets back 8. 8-7=1. One has won one unit. The corollary is this: One bets 2 to the power of n in the nth attempt. You have already bet 2 to the power of 0, 2 to the power of 1, 2 to the power of 2 etc. before. The sum of all 2 to the power of n is always one less than 2 to the power of (n+1). Proof by complete induction. However, I do not carry it out.
The second player also got lucky for a while in the example. His number came a little too often. He reached the peak with over 482 units, thus almost quintupling his capital (but please also look at the other diagrams)! But he is also exposed to the biggest swings. When his number comes up, there is always a big swing upwards. If it comes up a few times too many, he can easily get far ahead as well.
The rock-solid player who always plays a simple chance with a unit also came forward, but not as far as the others. He reduces his risk. However, this also does not put him in danger of winning much, his peak at 159 units. He also lives the longest. In total, his money has lasted 9773 trials, almost to the end of the trial series.
We just take a look at the expected values: The first player has the lowest negative equity. On average, he loses 1.35% of 10 euros per attempt, i.e. 13.5 cents. This results in an “expected lifetime” of 1000/0.135 = 7407 games. He even survived a longer time in the attempt. Player 2 operates with a 2.7% disadvantage. The result: broke after 4240 attempts. That is a good result. He would have to survive only 3704 games (7407/2) on average.
Player 3 has the biggest negative equity. He only plays with a 1.35% disadvantage, but he makes much higher turnovers due to his system. These naturally have a negative effect. His equity is the first to cross the zero point. But it is not possible to calculate exactly when it crosses the zero point. How often did he have to double? How long did he miss his chance before it came, if it came at all? These questions have a bearing on his equity. At the time he went broke, he should have had 16 units left. So he went bust a little too early. Although in this series of tests he had had the opportunity, through the initial successes, to be able to double even more often than previously planned.
As a side note: At the time he went bankrupt, he should still have had about 16 units. So he was a bit unlucky, or rather the long series of losses came a bit too early (if you take into account the 11 attempts, which are based on a chance of 11 coin tosses with the wrong side, you could calculate that the chance for this event is 1/2 to the 11th power, which is 1/2048. And that is 1/2048. So theoretically, the event “pair 11 times in a row” would have to occur after about 2048 attempts. But this calculation is not quite right either. Because the chance of impair is slightly less than 50%. And that already has an influence on many attempts. Besides, the player only had the chance to double so often because of the favourable previous course of events. At the beginning he could even have doubled only 5 times (10+20+40+80+160+320=630), so he would have played 6 times and would have gone broke against his chance the 7th time in a row (because he would still bet the remaining 370 Euros the 7th time).
Now a second run:
I always marvel at the diagrams myself the moment my computer spits them out. Then I first puzzle over whether I can expect such a result or whether I should rather run the simulation again so that “realistic” results come… But now I have made up my mind: I will always bring the next diagram and rather puzzle over what it means, together with you.
So the doubling player once again provides the most interesting result. He actually made it to over 3500 units. The crashes are always temporary. The “Impair” event always came just in time. But what is interesting here is that his equity has long been in the red, but he still has plenty of money. Is it all a coincidence? Or should one double, i.e. apply the system? But only under one condition: Stop in time, right? But that seems to be where the problems start: If you always knew when the peak was…
Above all, put in his position, you and everyone else would definitely think like this: “Gee, things are going well. I have so much money. Nothing can happen to me now.” Or something similar. Anyway, in the simulation, the collapse came after all. Bankruptcy after 8229 attempts. But it’s worth thinking about the first burglary. The sequence was such that the last time he bet he didn’t have enough money to double. His chance came, however, with the last bet. The last bet was only 429 units. So he bet again at the level of 429*2 = 858 (and this time he didn’t win the pre-calculated unit but still lost several thousand). But the second setback in a very short time, that was no longer bearable.
Player 1 and player 2 move rather unspectacularly here. Although it is astonishing that Player 1, the rock-solid one, even survived completely this time. But not exactly enviable: he still has 16.5 units. And played for so long for it. Player 2 was lucky again at the beginning and managed to get 582 units (why didn’t he stop? That stupid computer!), but then he was the first to lose everything. Broke after 5212 attempts (again, too long).
Now one last run. I promise I’ll take it as it comes (by the way, one run takes about 8 seconds, just to give you an idea of how stupid computers really are).
Here’s the result:
A promise is a promise. So I’ll get straight to the interpretation. These annoying bold straight lines have the following meaning: They represent the expected values. But I have programmed my computer in such a way that it calculates the expected values (or equity here) only on the basis of the stakes. And someone who has no more money doesn’t bet anything either. In plain English: two players were broke long before the time. So much too early. Does that bother us or not? No, why should it? They were “saving time”. The money would have been gone at some point anyway. In a playful sense, it was still bad luck.
The blue and purple lines represent (once again) player 1. He just plays solidly. Like a German. No indiscipline. His equity is gradually falling towards zero and so is he. You could also say “go for sure broke”. But he has briefly looked into the black a few times. At least. That’s not certain either. But I don’t want to make any more runs to prove it.
Player 2 and player 3 both had a higher peak than player 1 this time, but it didn’t help them. Player 2 simply didn’t get his number. I checked: It didn’t come up a total of 148 times. By the way, he had played the 30 (my computer does this randomly. But once he has decided, he sticks to it). The probability, just by the way, of a number not coming up 148 times in a row is 1.73% (needless to say, you know the method of calculation. But I’ll note anyway: 36/37 to the power of 148).
The fate of player 3 was simple: He had reached 184 units, then no impair 8 times in a row, one of which was zero, the 0. I’ll add a little arithmetic experiment here: How does the doubling system actually continue correctly when the zero comes?
The calculation is quite simple. So you had already doubled 2 times. That is, 1 unit lost, then 2 units lost, then 4 units lost. Now you bet 8 units. Now the zero comes. What do you do? First of all, take half of it off (that is guaranteed to be clever), that is four units. Let’s calculate in euros from now on, that’s better. So you lost 10+20+40 euros, all before the zero came. Now you have lost half of the 80 euros. But your goal with the doubling method is to win another unit, 10 euros, with every approach, i.e. after you have won a unit, a game. But now you have already lost 10+20+40= 110 euros. In order to actually win 10 euros again in total in the next game, if your chance comes, you must therefore bet 120 euros. 110 lost, sure, so you bet 120. And how much had you bet in the previous game? Yes, that was 80 euros. Now 120 euros. So you bet one and a half times that amount (so that you don’t have to do the same calculation again if the zero comes on the second, third or fifth try). After the zero, you have to come down (take half the loss) and bet one and a half times the previous amount.
Now I’ll put in two more illustrations because I find it so exciting myself. Please interpret them yourself:
The whole chapter serves above all to show how reliably mathematics works in the long term. Because: in the long term, there was no really successful career in any of the examples. More or less all the players went broke after a long time. Despite several temporary successes.
But I would like to point out: as reliably as mathematics works when you are at a disadvantage, it also works reliably for you and for everyone else when they are at an advantage. To be more precise: one side is also at an advantage in these examples. And this side has prevailed in the long run.