Roulette Paradox: applying the Parrondo paradox to roulette
If you are interested in learning more Parrondo's paradox, you must know that on December 23, 1999 the prestigious British scientific journal Nature, published an article that intrigued biologists, mathematicians, logicians and statisticians from all over the world.
The author is an Australian engineer, Derek Abbott, who illustrated the so-called Parrondo paradox.
The professor. Juan Manuel Rodriguez Parrondo is a physicist from the University of Madrid, who has shown how we can mathematically win by participating in two unfair gambling games (in each of which probability puts us at a disadvantage).
The Spanish physicist devised this application of his theories to competitive games to illustrate the methods of his research on the transport of proteins in cells, on certain peculiarities of the Brownian motion of the molecules of a fluid or gas and on certain problems of thermodynamics.
Parrondo's paradox describes two gambling games based on the toss of two coins.
Heads or tails, if the coins are not rigged (i.e. if the game is fair), the probability of winning is 50%.
In Parrondo's game A, the coin (which we will call coin X) is not balanced: on average it comes out heads only 495 times out of 1.000.
So playing game A, we certainly lose in the long run. In game B we still bet on heads, but we use 2 coins (to which we assign the names Y and Z).
Coin Z is very disadvantageous: it only gives heads 50 times out of 1.000 (one time out of 20); Coin Y, on the other hand, favors us, generating heads 700 times out of 1.000.
Another rule of game B is that we use coin Z only if we have a number of coins in our pocket exactly divisible by 3. If this number is not divisible by 3, we always use coin Y only.
Even in game B you lose in the long run, in fact the probability of winning is 1/3 (the percentage of times we use coin Z) multiplied 0,05 (i.e. one twentieth) which gives 0,01666 ... plus 2/3 multiplied by 0,7 which is worth 0,46666.
The sum of the 2 probabilities is 0,48333 or: less than 50%.
We conclude that we don't want to play either game A or game B.
Parrondo's demonstration is amazing, because if we play game A twice and game B twice and continue like this, or we randomly choose sometimes A and sometimes B, instead of losing, we win. The longer we play, the more we win!
Prof. Parrondo demonstrated this conclusion by resorting to rather sophisticated mathematical reasoning which is not the case to report now.
He also simulated various tranches of 50.000 plays on the computer, confirming this surprising result, which seems to definitely contradict our intuition.
The situation of the two games of chance described by Parrondo's paradox is formally identical to that of a harpoon that is rotated by a paddle wheel, moved by the molecules of a gas that hit it randomly.
A harpoon is a toothed wheel with teeth inclined like those of a saw, which can turn in only one of the two directions, that is, not in the opposite direction, because there is a harpoon that sticks in the hollow between two teeth and blocks the wheel (it's the 'a' element in the figure below).
In the permitted direction, however, the harpoon slides on the upper surface of the teeth and does not hinder rotation.
Ideally such a structure could take energy from the gas molecules that go randomly in the right direction and be insensitive to those that go in the opposite direction.
At first glance it might seem that this device is capable of violating the Second Principle of Thermodynamics, because it would draw energy from a gas at a single temperature, without exploiting a jump from hot to cold.
Of course this is not the case, it is not possible to violate the Second Principle of Thermodynamics and Parrondo's subtle reasonings, born to explain the complex mechanisms of nature, will only be useful for those who really make the effort to understand them.
Application to roulette
Many years have passed since this fascinating theory was first presented, arousing the interest of the academic world and the entire scientific community.
Over the years there have been countless attempts to apply it to the green table, but at least that I know of no one has so far succeeded.
Because the paradox is winning
Analyzing the implantation of the Parrondo paradox, I was able to deduce that it is successful as the ABBAB game matrix in practice 'dilutes' the game frequency of coin Z (the one that comes out 50 times out of 1.000) just enough to ensure that the best coin (coin Y + 70%) can win an amount greater than the sum of the losses of coins X (win frequency 49,5%) and Z (5%).
In practice, only in game B of Parrondo's paradox, coin Z, is played only when the cash is divisible by 3, so it is used 1/3 of the time, or about 33,33%.
However, by including game A in the context, if the scheme to be followed for choosing the coin to use (which from now on we will call 'matrix') is always ABBAB, this will be played 40% of the time (there are 2 terms A out of 5 in the matrix), so that coin Z from its 33,33% frequency in game B alone, drops to 1/3 of the remaining 60%, or 20% in the global game, i.e. just enough to allow to coin Y (+ 70%) to be able to overcome the disadvantage caused by the other two adverse coins.
In essence, Game A, although disadvantageous in itself, acts as a nuisance (noise) on the most disadvantageous component (coin Z) of game B.
Because the paradox is not winning
The apparently insurmountable problem of roulette is that there is no coin (bet) that has a 70% probability of sortie and that, in the event of a win / loss, only counts 1 unit as a real coin does.
It is also true that if this coin existed for roulette, it would be enough to play it directly and, as you will imagine, all this writing of mine would make no sense.
However, the original system of the paradox has been amply demonstrated to be successful, that is, it is a probability interlocking such that, assuming the coin Y really exists, it mathematically exceeds the house advantage.
Our aim therefore, if we want to try to apply it to roulette, must be to reproduce it in a 'faithful' way, leaving the task of recovering the cost of our artifice to recreate the Y coin (+ 70%) to a maneuver.
Before venturing into the creation of the artifices, however, we must establish a very important thing, that is to understand how much our system wins as a percentage, in order to learn what its yield is, a fact that will serve us to determine a basic element of our game, or ...
The great misunderstood: the Stopwin
In the field of play, it has always been believed that Stopwin is useless, since, if a system wins, it always wins, so Stopwin has no sense other than to shorten the game unnecessarily and the same applies if the system loses. The Stopwin, as I understand it, is fundamental.
Starting from the certainty that at the moment there is still no mathematically winning system at roulette, the Stopwin becomes decisive when used with awareness.
I often read on the various online forums, posts by players who say they apply a Stopwin of 3/4/5/10 units per session with their method, but without ever specifying why they chose that number of units.
Perhaps we tend to reason that maybe winning 5 units a day of 10 euros, at the end of the month they make 1.500 euros, practically an extra salary, but in this case I agree with the majority of the players, the Stopwin makes no sense, at least as long as it remains a number of subjective entity.
The Stopwin, to have a concrete usefulness, must first be quantified and this can only be done by verifying a priori what is the yield of the method that we are about to apply to the green table.
Example: if I have decided to set up my game in such a way that it yields 10% on average and I decide to play 100 shots per session and, after playing only 70, I already have the 10 units in hand, well, besides being lucky, I collected 3 units that were not due to me, because my system, having an average expected yield of 10%, out of 70 strokes played had to make me win only 7 and not 10.
Having reached my goal (10 units in 100 shots) with 30 shots early, I stop, I apply a Stopwin, as given the yield of the game I am applying, in the remaining 30 shots I am more likely to remain in the ballot or worse than lose, while exposing myself to the tax without any reasonable reason.
Another certainly valid reason for applying a Stopwin is linked to what I empirically demonstrated a few years ago through a PC simulation, where I ascertained that Marigny De Grilleau was wrong, Because personal permanence does not exist or better:
in roulette, scraps are analyzed only in reference to the generator that produced them in an uninterrupted series of spins.
Therefore, if the aforementioned roulette has made me win more than the theoretical due, I apply a Stopwin and this does not mean that I no longer play that day, but that that day I no longer play that roulette.
The uselessness of the Stopwin is the biggest lie that the player tells himself to avoid having to stop playing, it is a further gift we give to the dealer, because if a wheel has favored us, by not stopping we allow it to recover the scraps (gap) that he just produced in our favor.
Roulette Paradox Simulator
To establish what the yield (roi%) of the original Parrondo paradox is, I created a simulator that exactly reproduces the 3 coins system and thanks to which I was able to analyze the succession of events inherent in the dynamics of this theory, as well as obviously try also other always winning variants, thanks to the different possible mixes (matrix) of games A and B.
In the Paradox Simulator, in addition to the game matrix, it is also possible to vary the percentages of the 3 coins, so, without prejudice to the fixed ABBAB scheme (matrix), it is also possible to try to replace the percentages of the original coins, with percentages corresponding to those of the bets that can be made at the roulette table.
Before doing this, however, in order to determine the roi% of the Parrondo paradox, we must identify with certainty what will be our 3 coins, which for simplicity we will continue to call X, Y, and Z.
The three coins
Starting from the sortie percentage of the 3 coins used in the Parrondo paradox, let's now try to understand what Roulette can offer us.
Calculating the percentages to be entered in the Simulator is very simple: just divide the chance we want to consider by 37; for example, a simple chance is made up of 18 numbers, so 18/37 = 0,4865 so in the program we will insert the value 486.
As I have already said, the coin X that in the paradox wins 49,5% of the time for us can easily be a simple chance, which wins 48,65% of the time (it is not necessary to split the percentages to the decimal).
This first coin is really perfect, because in fact it only wins or loses one unit, just like a real coin.
Coin Y, which in paradox wins 70% of the time is our 'driving force', it is it that generates the profit, and it is it that unfortunately, not being immediately available, we have to recreate with an artifice.
Remember that to reproduce it we must try, as much as possible, to simulate the condition of a real coin; you can have fun as you see fit, in my experiment I considered a simple 2 terms martingale, as this wins 1 unit with a statistical frequency of 73,63% and when I win I get 1 unit (perfect!), but when I lose unfortunately, I lose 3 units instead of just one (disaster!).
These lost units will therefore be the object of our recovery attempt, a task that is delegated to the Z coin (we will see how soon).
The coin Z, which in the paradox has a frequency of 5%, can be reproduced very well by betting a horse (split), which at each spin has a probability of sortie of 5,41%, absolutely in line with the percentages of the paradox but this when coin wins, unlike that of the paradox, it does not win just 1 unit, but it wins as many as 17 (excellent!).
Let's stop for a moment before continuing; now we have identified the exact sortie percentages of our 3 new coins which are:
- Coin X - 48,65% - simple chance;
- Coin Y - 73,63% - 2-term martingale;
- Coin Z - 5,41% - horse (split).
At this point we have all the elements to establish, thanks to the Paradox Simulator, how much our 3 coins would win if coin Y lost only 1 unit instead of 3 and if coin Z won only one unit instead of 17.
Determine the yield (roi%)
If I enter the three percentages in the Simulator, that is 486 for the coin X, 736 for the Y and 54 for the Z and run a million simulations, it will result that this system has a roi% of about 3,5%.
At this point, having identified the 3 coins, I have to organize myself to manage the bets in order to try to recover the 2 units that coin Y loses more than a real coin, as if I succeed in this undertaking, in the medium / in the long run I can only make sure I win 3,5% or about 3 and a half units every 100 strokes played, fee paid of course!
Let's continue: finally we have the 3 coins and we know exactly in which sequence (matrix) to place the bets:
GAME A - Coin X (one simple chance)
GAME B - Coin Y (martingale of 2 terms with stop at the first unit won) or Coin Z (split) only if the cash is divisible by 3
GAME B - Coin Y (martingale of 2 terms with stop at the first unit won) or Coin Z (split) only if the cash is divisible by 3
GAME A - Coin X (one simple chance)
GAME B - Coin Y (martingale of 2 terms with stop at the first unit won) or Coin Z (split) only if the cash is divisible by 3
From the sixth shot onwards, it starts again from shot 1 and so on.
I play randomly and win!
How to choose the chances to bet? I have decided to choose them randomly and I will immediately explain why.
Mathematicians for centuries have admonished the poor player with concepts such as 'at roulette every stroke is a new shot' or 'roulette has no memory' well, since I agree with this assumption, in the new program that I have created to manage the cash and bets (I'll talk about it shortly), I inserted a random generator that at each shot shows me what I will have to bet based on the type of coin I have to play to follow the ABBAB matrix.
This aspect of random play in my opinion is as important as everything else, because when I sit down at the table, I have no idea of the scraps that specific roulette has produced in the past.
In fact, if, for example, I decide to always play the X coin in red and that roulette the day before had gone strongly in difference with the red one, I will probably suffer all the negative wave of black, while the exact opposite is also true and therefore I might as well win, so not being able to predict why each shot is a new shot, then I entrust my fate to the random generator of the program, in order to have a 100% method based on the percentage of future events and never on past ones, which as far as I know have always been a source of defeat for the player.
So, in a nutshell:
- if I have to play coin X: I play a simple chance at random;
- if I have to play coin Y: on the first shot I play 1 unit on a simple random chance (if I win stop), if I lose, I play 2 units on another random chance that is not necessarily the same as that of the first shot, in how much our% of victory does not change if we change chance because it always refers to the possibility of guessing 18 numbers out of 37 in two consecutive hits;
- if I have to play coin Z: a horse (split) at random among all those available on the playing mat.
Manage the cash (Money Management)
We now come to a really important topic: the management of the cash register and the recovery maneuver.
We remind you that, thanks to the simulator, we have verified that if coin Y did not lose 3 units and if coin Z did not win 17, it would have a yield of 3,5% (let's never forget this value); however, in order to faithfully apply the paradox scheme and in order to allow it to correctly indicate the bet to be made, we need to create a dummy case, which we will call Paradox case.
This box will be used to understand the trend of the paradox surrender and allow us to make the necessary considerations during our attack.
The Paradox Chest, therefore, will be accounted for exactly as if each hit / coin produces +1 or -1 thus excluding both the extra units lost from coin Y, and those extra won from coin Z.
This cash, if accounted for in this way, fully respects the paradox scheme and therefore can only win mathematically in the long run.
The second Checkout we need, it is that instead Reale, where we are going to account for exactly how much we have in our pocket compared to the spins played (real roi%).
Last but not least, the third speaker, which we will call Recovery box and in which all the extra units lost by coin Y and those won by coin Z will converge.
So if I win a shot with the:
- Coin X: +1 sign in the Royal Chest and +1 in the Paradox Chest;
- Coin Y (whether I win on the first or second stroke of the martingale): +1 sign in the Royal Bank and +1 in the Paradox Bank;
- Coin Z: +17 in the Royal Chest and +1 in the Paradox Chest and +16 in the Recovery Chest.
On the contrary, if I lose with the:
- Coin X: -1 sign in the Royal Bank and -1 in the Paradox Bank;
- Coin Y (if I also lose the second shot of the martingale): sign -3 in Cassa Reale, -1 in Cassa Paradox and -2 in Cassa Recovery;
- Coin Z: -1 sign in the Royal Bank and -1 in the Paradox Bank.
Why in Cash Recovery if I win with coin Z +16 instead of +17?
I do it because one unit of the 17 won always goes to coin Z, as if it were a real coin, the remaining 16 constitute my 'surplus'; always remember that the Paradox Cashier must be managed exactly as if the game were carried out with three real coins, which only provide +1 / -1.
Therefore, if during the game I have to play the coin Z (split) this I do it only if the Cash Recovery is in negative; on the contrary, if the Cash Recovery is zero, the coin Z does not need to recover anything, why penalize yourself with a bet at 5,41%?
In this case I play my best possible coin, which is a simple chance instead of a horse.
At this point, however, if every time the Z coin is played, the Recovery Case is at zero and I play a simple chance, the yield% of the system is no more than 3,5%, in fact using the Simulator and substituting for the Z coin the value 54 (horse) with 486 (simple chance) will have that the yield from 3,5% rises to about 18%, a value that will identify our RMP (maximum possible yield).
Drain the overdraft
The Recovery Fund can also be literally absorbed by the Royal Bank, how?
For example, if after 10 hits it went particularly well (always remember that coin Y travels at 73,65% probability) and in the Royal Bank I have for example 4 units, these 4 units are an enormous amount as Roi% compared to spins played (40%) and since the maximum yield we have verified to be 18% (let's say 20% to round up), I have 2 units that are not due to me and which therefore I 'move' accounting from the Real Cash to the Recovery one, allowing me so one thing really important: delay the recovery maneuver by one term with the horse, because when coin Y loses 3 units, the two that go to Cassa Recovery will be totally absorbed by the previous surplus, allowing me to play coin Z again with a simple chance instead of using a horse.
The same goes for the case in which the Recovery Cash for example is at -8 and I win the bet with the horse; of the 17 units won 1 goes to the Paradox Box (mandatory) and the remaining 16 bring the balance of the Recovery Box from -8 to +8, which means that for 4 spins lost with coin Y (remember that this coin generates -2 units for each loss), I will be able to avoid starting the maneuver with the horse.
Finally, considering the minimum expected Roi (3,5%) and the maximum possible one (18%), I believe it is appropriate that all the calculations relating to the allocation of the units of the Royal Bank and of the Recovery one, but above all to the Stopwin, should be calibrated on a profit of 10%, so if I have a 10% roi on spins played, I stop (Stopwin) and start another session (clearing all calculations) on another roulette.
Well folks, actually I would also have more to add, but if in the future someone were to ask you for info on the Parrondo Paradox applied to Roulette, you know where to direct it.
Also remember that what I have presented here is only one of the many possible variants that can be created and that, thanks to the simulator, you can test for yourself; Finally, I would like to point out, if by chance it had escaped someone, that in practice we play on an even mass!
With this roulette software we close the discussion of this hopefully interesting topic, or the famous Parrondo Paradox.
What I am presenting to you now is in summary an accounting tool (money management) based on this mathematical paradox, to which I have however made some small changes that I will illustrate more later.
The new gaming facility of Roulette Paradox
As already mentioned, I have made some variations to the original system, as as I hope it is now clear to everyone, the problem in recreating the Parrondo paradox for roulette consists solely in the impossibility of having a coin that wins about 70% of the time and that only 1 unit wins or loses, as a real coin does when playing heads or tails.
To overcome this obstacle, the only way is to create a fictitious coin (chance) that has that percentage of sortie, but which in the event of a loss triggers a recovery maneuver to bring the lost unit back to the cash register in addition to a classic one. currency.
Our 'rigged' currency in this maneuver is always the Y currency, except that I thought of using a less expository combination than the one described above, which therefore makes the necessary and inevitable recovery phase less difficult.
I have identified this coin in betting with a single shot 2 single dozen.
In this way we have a fictitious coin that when it wins (64,86% of the time), it collects one unit (and this is perfect) and when it loses it makes it necessary to recover only 1 unit (in reality, 2 are lost, but 1 is what would lose a classic coin, so only one unit needs to be recovered).
For the other 2 coins the situation is less complex, we will aim for a simple chance at random to our liking.
We insert in the Paradox Simulator the percentages of sortie of these three coins, that is:
- Coin X: 486 / 1.000 (simple chance);
- Coin Y: 648 / 1.000 (2 dozen or 2 columns);
- Coin Z: 486 / 1.000 (simple chance).
Thus simulating the paradox for a high number of spins, we will first verify a very important datum: the yield of the system in the absence of artifices, which is approximately 10%.
I have long evaluated the choice of this setting, because as you can see from the previous graph, a fictitious yield of + 10% definitely sets us apart from an 'excessive' variance, which is the worst killer immediately after the tax, because it is higher. is the negative variance (which is also present in a game with positive EV) and the greater the psychological pressure we have to bear in the negative game phases.
Suffice it to say that a mathematically winning system with Roi + 1% or + 2%, could go through negative cash phases of several hundreds if not thousands of hits before returning to the positive, would we all be able to withstand such psychological pressure?
Use of Roulette Paradox
The use of the program is quite simple, we start by betting one unit on the simple chance indicated in the box below (BET ON BLACK in the figure) and updating the cash box, by pressing the buttons on the side in case of a win (green check) or loss (red cross).
At this point, just click on the next coin (Y / Z) based on the one of the two that will be activated automatically and therefore point the chance indicated by the random generator of the software.
As a unit you must bet the amount indicated in the yellow box at the bottom left (Bet Units), for the X / Z coins this will always be equal to 1 unit (even mass), for the Y coin (two dozen) it will be necessary to bet the amount indicated on the single dozen, so if '3' appears in the yellow box, they must be bet 3 units on the first dozen and 3 units on the second dozen, or those that the program will indicate.
Once you get to the last coin on the right (the fifth), you start from the first coin X and so on.
Doing some tests with real stays that you will surely find on the net, you will see that the Paradox Cash will always score +1 or -1, exactly as foreseen by the Parrondo paradox, while the General Cash and that of the Session in progress (Current Attack) if yes loses with coin Y, they will actually score -2 units (if the bet was 1 unit per dozen).
As everyone can verify thanks to the Simulator, a system set up in this way generates a fictitious roi of + 10% and this is mathematical, the Paradox Cash in the medium / long term will 'inevitably' align with this value, but not for who knows what magic, but simply because when he loses with coin Y he will only score -1 instead of -2.
Our aim will therefore be to activate a maneuver to recover the extra units lost from coin Y.
The simplest thing would be to play a 2 dozen progression, with an increasing bet 1/3/9/27/54 ... units per dozen, but then we would go bankrupt within a few minutes and this is certainly not our goal.
I therefore thought of protecting the recovery in this way: first of all if the fictitious yield of the Parrondo paradox using these 3 coins is 10%, in reality we will try to snatch about half from the bank, or a more than decent 5% of Roi in Real cash, which is already a business.
If in the course of the single session the X / Z coins favored us in terms of winning spins compared to those played, these extra units will serve to slow down the increase of the bet.
For example, if after 20 spins I have +4 units in cash, it means that I have 3 more units compared to 5% on the shots played (in fact I should be at +1), well, if at this point I lose a spin with the Y coin (which on average wins about 2 times out of 3), the software does not immediately raise the stake for recovery, but remains at stake 1 until the Roi of the real cashier eventually falls below 5%.
This means that in the example above I can sustain another 2 losing hits on Y without necessarily having to raise the bet.
The second trick to keep the stake low consists in the fact that the recovery maneuver is set only on coin Y, this helps because the expected win frequency at each stroke is about 65%, much higher than that of the other chances played. with X / Z coins (48,6%) and consequently also the negative phases will be less long, even if the loss in this case is double, but you can't have everything!
Contain the bet with Roulette Paradox
The third factor of containment of the bet consists in the fact that having the coin Y a probability of 65%, that is to say winning on average 2 out of 3 hits, the software will attempt the recovery not in a single hit, forcing us almost to a tripled martingale, but rather always in 2 strokes diluted to infinity, result obtained by always dividing the recovery by 2 at each attempt.
How is the recovery calculated? It is not simply the number of units lost by coin Y, but it is about 50% of the difference between the Paradox and the Real Cache.
In fact, if at a certain point in the session the Paradox Cashier (which I remind you will record 10% profit on the strokes played) is at +6 and the Real Cashier is instead at -4 due to the units lost on the Y coin, the recoverò will be set by dividing by 2 the difference between +6 and -4, that is 10 units, so the stake in this case will be 5 units per single dozen.
In case of victory of the shot (which we remember has a success rate of 65%), the new balance will be +7 for the Paradox Cashier and +1 for the Real one and therefore the new recovery bet will be equal to 3 units per dozen.
In the event of a loss, on the other hand, the Royal Bank will go to -14 and the Paradox to +5, so the next bet on coin Y will presumably be 10 units per dozen which, as you can see, is only doubled and not tripled as it normally would happen with a recovery progression in two dozen.
Furthermore: since coin Y, according to the classic rules of the Parrondo paradox, is played only if at the time of the bet the cash is not divisible by 3 (in this case, in fact, coin Z is played), especially when we are at roi levels. in line with expectations, it will happen that the positive variance of the X / Z coins will lead to delaying the trigger of the bet increase, which in fact is not necessary if we lose a few strokes with the Y coin but the roi is still in line with the goal of + 5%.
In Roulette Paradox it's all automated, just point what is indicated where indicated and click on the buttons to record the result of the shot (won / lost).
The program also features a function to save session data and a graph to check the progress of the Real Bank (buttons at the top right).
Furthermore, as long as the roi% of the Real Cash is 5% or higher, the green button (Target Roi) with the inscription 5% flashes, this to signal us to 'keep calm', because the red button (Next Bet on Coin Y ) will instead mark how much we should bet on coin Y (value to always multiply by 2, since in fact we must bet on 2 dozen) and if the indicated bet is for example 7 units per dozen, but in any case our real roi is 5% , who forces us to raise the bet? Just select '1' in the yellow box Bet Units and lower the proposed value.
Remember: the software 'suggests', but when in doubt we are free to decide based on the real situation of our cash register.
We come now to the usual questions: but what advantage does this system bring to the player? How much more capital does it take? Should a Stopwin or Stoploss be applied to the sessions?
The maneuver described here cannot bring a mathematical advantage to the player, because the expectation on the probability of winning at each spin is not changed, what must instead be verified is the distribution of the discards, which in this system would seem quite punished by the fictitious expectation of winning the Paradox (+ 10%).
At the time of writing this article I have played about 2.000 real spins (online roulette strictly with live dealers) and I must say that I have never deviated from the expected value (5% real) except with fluctuations that are all in all contained.
I know that 2.000 spins are few, however, as the statistics predict, I also encountered 6/7 consecutive losing hits with the Y coin, which with a normal recovery upright for 2 dozen would have produced a considerable overdraft, instead the highest bet that I had to do so far was 30 units per dozen (total 60 units), a phase which however quickly returned precisely because I remind you that every shot Y wins on average 65% of the time.
Based on my tests I would say 400 units are fair.
If, as we will try to verify, this maneuver 'softens' the variance without pretending to subvert the laws of the case, an initial capital of just 100 euros would be enough, which corresponds precisely to 400 units of 0,25 cents.
Well, this is almost everything, obviously I recommend that you always do a lot of tests with the numerous real stays that can be downloaded on the net before putting a single cent on the green carpet (real or virtual), trying is free and above all it allows you to really test what we can expect during the real game in terms of negative phases and maximum bet, moreover nothing prevents someone from thinking of some tricks to improve the resistance of this selection method to apply the famous Parrondo's paradox to roulette.