Roulette Ideas

One of the biggest issues the gambler and indeed an investor faces is dealing with variance.

Anyone who understands the investment dilemma also understands that even with a positive growth expectation the result is still ruin, due to variance.

Quite simply an investment return or loss of 10% with equal/fair probability never gets to even as 1.1 × 0.9 = 0.99 (which is always lower than what the investor started with).


Let's consider a simple example using 25% to make it more obvious, where the risk taker nets 25% of their investment half the time and loses 25% the other half of the time. This is a fair game meaning it's better than any casinos.

So with an initial risk of $100 the gambler/investor/risk-taker nets $125 on a win, or has $75 returned on a loss.

So let's look at just 2 bets, starting with $100 and fully investing the balance each time vs the possibilities:
WW $100 → $125 →$156.25 (+$56.25)
WL $100 → $125 →$93.75 (-$6.25)
LW $100 → $75 → $93.75 (-$6.25)
LL $100 → $75 →$56.25 (-43.75)

We can see that the result is still a zero sum on average when we add up all the numbers in brackets, confirming that the bet is "fair".

However the gambler/investor does not get the average return! What the gambler/investor receives is known as the geometric return.  Quite clearly there is only one scenario out of 4 where the gambler/investor does not lose money even though it is a fair game!  As time progresses the volatility will more often than not reduce the balance to zero.

Hence why ruin is guaranteed just from variance alone, even with a fair game.

What hope then does the gambler have with a negative expectation game if a fair game cannot win?

Does the secret not then lie with converting variance into growth through processes known as volatility harvesting?

Something to consider is the famers fable, which visually explains the concept of harvesting volatility and the train in nature and socially cooperation beats selfishness (which is an emerging economic theory).

Exactly but we have from our side money management "that help us " in my plan always is allowed to have some losing session , but in the end of the plan monthly weekly I'm in positive also with some losing session

Here is a simple example of taming variance, or even better, converting it into growth.

Using my previous example, with a bet/investment that either wins 25% or loses 25% half of the time.

If you stake half of your money, each on two independent coups/sessions, then rebalance, the ratio of win to loss remains the same, but you end up with a new outcome where a win and a loss cancel leaving a zero net change. This new zero state has effectively tamed variance and converted it to growth.

The possible outcomes for this split allocation of $50 each using the same return (win or lose 25%) as per my first post. The possible outcomes are:
W and W  $62.50 + $62.50 = $125 (+$25)
W and L $62.50 + $37.5 = $100 ($0)
L and W $37.5 + $62.50 = $100 ($0)
L and L $37.50 + $37.50 = $75 (-$25)

For this single coup as can be seen the win and the loss remains the same, but we have 2 outcomes where we get a zero return.

25% of the time we win, 25% of the time we lose and 50% of the time there is no change. We manifested a new zero return state out of thin air, with an outcome of "no change" being reduced volatility.

If we just made a single bet our outcome would be $125, 50% of the time and $75 the other 50% of the time.

Clearly volatility is reduced with the split bet approach however the ratio of win to loss remained the same at ±$25 and the total of the outcomes is still a zero sum game.

Now let's play two coups with our $100 back to back as per my first post and see what happens with this rebalancing method with 3 possible outcomes of (Win, Even, Lose):

WW (1/16) $100 →$125→$156.25 (+56.25)
WE (1/8) $100 →$125 → $125 (+$25)
WL (1/16) $100 → $125 →$93.75 (-$6.25)
EW (1/8) $100 → $100 → $125 (+$25)
EE (1/4) $100 → $100 → $100 ($0)
EL (1/8) $100 → $100 → $75 (-$25)
LW (1/16) $100 → $75 → $93.75 ((-$6.25))
LE (1/8) $100 → $75 → $75 (-$25)
LL (1/16) $100 → $75 → $56.25  (-$43.75)

Again note that the ratio of win to loss remains the same but now our probability of being behind has been reduced to (1/16 + 1/8 + 1/16 + 1/8 + 1/16) = (7/16) = 43.75%

So now we have a 56.25% chance of being even or ahead. That right there has converted volatility into growth.

Contrast the above when playing two coups without splitting and rebalancing, your probability of being behind is 75%, or said another way the probability of being even or ahead is just 25% with a fair game.

So we can see that as time goes on, volatility reduces growth, whilst reducing volatility is converted into growth because it improves the geometric return (due to softer down swings). The gambler/investor never gets the average return (also known as the arithmetic return) you are always growing or shrinking your balance  coup to coup or session to session, the average return is only a theoretical concept.

This is not something new, it was originally discovered by Claude Shannon (a genius on information theory) and also a collaborator with Edward Thorpe.

The above is best applied across sessions.

If you split your overall bankroll risking less per session you reduce the inter-session volatility and that translates into growth (if you have an edge) because bankroll grows or shrinks geometrically, and volatility hurts growth (even with a substantial edge, ruin is almost certain).

Pretend you have a team of players and divide your capital across multiple session bankrolls. Continually pool the session totals each round and split your capital evenly for the next round. This will reduce volatility on your overall bankroll and translate to growth so that over time your session bankrolls increase. Introduce more session bankrolls "players" as you grow.

As you grow this in turn means you can play at a higher unit level which then compounds the winnings further, just like in the farmers fable, one player losing doesn't hurt the growth. The session bankroll increases and the number of session bankrolls also increases and this means each session is a smaller risk in percentage terms vs your overall capital.

The net effect is you pay lost sessions from compounded wins. But the compounding effect will not work if you don't control volatility, just like the farmer.

Now I have shared the method for doing exactly this, and it is more important than the actual method of play, because without it, just like the famer who even with a positive growth expectation you will most likely still lose because of volatility.

Beat the casino by controlling volatility of your capital, turn chance into growth, and pay the piper from compounded winnings.

Focus on developing a reliable method where session loss events are as rare as you can make them, number of units required is modest and profitability is sufficient to replace a bankroll in 5-10 sessions.

The rest is letting the magic of volatily harvesting and compounding do its thing.

This is a new frontier in economic theory (current theory mostly has invalid assumptions) and it is based on principles of cooperation and sharing in nature for long term survival.

Why this behaviour manifests in society and nature is that it beats purely selfish behaviour due to the fact that one or two negative events can wipe you out along the way (life is a non ergodic procces) and you never get to realise the "average" result or return.

We instead get a time dependent return in gambling and investment (known as the geometric return), step by step our fortunes and growth is crushed by negative events. Digging out of holes gets less likely as another negative event is just around the corner (as I showed 76% probable to be behind in just two bets).

So instead, use volatility harvesting and settle for more break even and modest returns than more losing and harvest the good times when they happen.

Here is quite a nice article to enlighten one on Russian roulette, gambling and investment returns and the concept of ergodicity. Thoroughly  recommended reading:

https://taylorpearson.me/ergodicity/

The harvesting as I have described works by reducing volatility because it introduces a break even state whilst the ratio of wins to losses remains the same.

Geometric return is typically going to be negative if you don't manage the volatility (even when investing with a positive expectation, refer to farmers fable to see why).

On a regular EC bet the volatility creates an effective 12.5% drag which massively eclipses the house edge that most people are worried about.

To be clear the win loss ratio staying the same means the house edge is still alive and well and is not altered, the negative expectation is still present.

When we combine the splitting and rebalancing back to back we end up with 9 possible outcomes where breaking even dominates and smaller wins and losses is the norm whilst extreme movements is the exception. This is with just 2 virtual players but we can have many more.

If your method for playing a session is robust, you should have far more winning sessions than losing with sufficient probability to replace a bankroll in 5 sessions. If you are having to use many hundreds or thousands of units you are most probably not on the right path and need to figure that out first and come back to this topic later.

So working on the basis that you have a method that provides gains with sufficient reliability (distance between loss events) then harvesting volatility should be beneficial, you will have more net winning states than losing and both the extreme winning and the losing states will be far less likely.

After 3 rounds we have 27 states (again with just two virtual players) and the extreme events are even further on the fringes and the probability of being behind is reduced even more. Four rounds and we have 81 states with the extreme outcomes pushed even further to fringes.

The other important point is to also to adjust the session/coup unit size growth/contraction in line with net balance growth or contraction. This how optimising the geometric return is achieved.

You may find that you can't bet a casino denomination. So adjust number of players to align as closely as possible to a denomination or just keep some in cash reserves for that round, or both.

Let's say a round requires 10 units and you are playing with a $500 bankroll. Let's say min bet is $10 meaning each player requires $100. So $500 divided by $100 is 5 players.

Let's say your first round you have new net balance of $620. Now you have 2 options if you stay at the same unit size you can play 6 players at $10 units ($100 each) leaving $20 in cash which is not at risk, or if you increase your unit size to $15 you can play 4 players again leaving $20 in cash not at risk. Playing less players does increase your probabilty of ruin so be aware of what you're trading when you reduce the player count. (More on this below).

Let's say you had a bad round and are now down to $460. Again you need to play a minimum unit size of $10 and so dividing by $100 we get 4.6 players or if we stay at $15 unit we can use $460/(15×10)  = 3.06 or 3 players. Given we want to stay close to 5 to reduce risk of ruin we expect to play 4 players.

Let's say we therefore reduce our unit size to $10 units and use 4 players, leaving $60 in cash not at risk.

That's the basic idea of adjusting number of players to work with the casino denominations and the available balance. I would try to not go much below the number of players that meets long term survival requirements as we don't want to reduce the risk of ruin in a round too much below a 1 in a billion event. The number of virtual players required will depend on your round robustness or failure rate so you need to work that out using math not just guessing.

It's usually not that difficult if you know the sequence  of events that lead to a win then multiply those probabilities together and subtract this from 1.

Or you can work it out from the other side and list the sequence events that lead to a loss in that round, just multiply those probabilities together and that's your probabilty of failure.

If you have a complicated betting method with lots of conditional logic or loops then you may need to use a Markov model to compute your probabilty of winning or losing (known as the absorbing states).

Once you have your probabilty of loss then you just need to multiply that by itself (raise to a power) until you get a number that is less than 1 in a billion.

Example: let's say you bet a dozen using a the sequence 1 1 2 3. Which needs 7 units. It has a probability to fail of (25/37) ^4 = 0.2084265973 which is ridiculously high.

This is roughly 20% or 1 in 5, which is a long way from playable system but still useful for this exercise.

Ok so we can experiment on our calculator to find out how many back to back failures need to occur before we lose our entire bankroll in a single round so that the probability of this event is less than 1 in a billion.

1÷((0.2084265973)^(5)) ≈ 2,542
1÷((0.2084265973)^(10)) ≈ 6,463,512
1÷((0.2084265973)^(13)) ≈ 713,853,675
1÷((0.2084265973)^(14)) ≈ 3,424,964,400
1÷((0.2084265973)^(15)) ≈ 16,432,472,846

So we see that the magic number of virtual players/sessions needed is 14 but 13 isn't too bad either and 15 might divide better into casino unit sizes and give additional resilience to 1 in 16 billion.

A faster way to calculate the number of players/sessions needed is rather than the guesswork above we can use the log function on our calculator:

log(1,000,000,000)÷log(0.2084265973) = -13.21

So we can ignore the minus sign and round UP to a whole number, which is 14 players per round. If we play with 14 players we will only have a total loss where every player loses in the round greater than 1 in a billion. You still may have some substantial losses due to the bad method, but not a single round total loss.

Personally I would not use a martingale/Fibonacci  progression on the dozen, so please understand this was used solely for illustrative purposes only and is not a recommendation. You will lose just as easily as a martingale and sometimes worse due to the proximity requirement of multiple hits that a Fibonacci progression requires.