Okay, so I thought I'd contribute a little bit of wisdom or at least food for thought.

What would you say, if I told you that you can make money on a mutually exclusive, binomial event such as flipping a coin (50% probability)?

It blows most peoples minds when I tell them this. Here are the conditions......A) You must have control over the wager amount, B) You must have a significant starting capital (say 100x the initial wager amount) and C) You must adhere to a strict diet of progressive betting strategy.

Here's how it works: Progressive betting requires that it be truly mutually exclusive and also requires inherent "strings" be present over a given sample of n events.

Take for instance, our coin flip example. If you have a data sampling of say 10,000 flips. Over the course of that entire data set, you will experience both positive and negative strings. The distribution will follow an exponential decay (i.e. there will be a very small number of long strings and there will be a very large number of short strings.) A typical distribution for strings might look like this......10 strings 1, 9 strings 3, 8 strings 7, 7 strings 15, etc.

Ever wonder why typical profit analysis tools show strings? Here's how to make it work.

If the event is truly mutually exclusive and binomial, over large samplings, you'd expect the number of winning strings to be similar (negligibly identical) to losing strings.

So, you bet initally 10 units. If the outcome is a win, you take 50% of the profit and leave the other half out as an additional wager increase. If the next outcome is positive, you do the same, take 50% and leave the rest, again upping your wager. When you lose, you restart the wager sequence back to the original wager/lowest increment.

So over a string of 5 wins (10 unit start bet)....it would yield 65.9375 units of profit and the same losing string would yield a loss of 50 units.

The key to progressive betting is inherent strings. The system fails and you actually lose money if there are no strings and it's simply a binary oscillator (i.e. win, loss, win, loss, win, loss, etc).

This is exactly how successful blackjack betting strategies are employed.

The second part of this segment is second order effects and how even with a 50% probability and a "player" using progressive betting will still end up losing.

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So the casinos are basically a license to print money. They have several strategies for reducing already less than 50/50 odds. (there's NO game in a casino with an equal or higher than 50/50 odds.....blackjack, when isolated and taking advantage of the double down, betting feature can yield 51/49 odds in your favor, but that's a topic for a different session).

Here's how the casinos REALLY make their money.

Second order effects ensure your doom (in the long run).

Basically, the house is an infinite and unlimited bank. This is a good assumption for all but the most rich "whales" who walk through the door. The casino knows, that even if you have "even" odds, all they have to do is get you to play long enough and they'll get you on a negative string long enough to send you back to the atm. The house on the other hand, can endure virtually limitless strings of positive gain on your behalf. This is exactly why they treat whales and high rollers differently. Those players have a much higher ratio with respect to bankroll to the house, so they can go on a positive run and really hurt the casino. This is why they do everything they can to get the player to continue playing and dump it all back.

So as stated above, a condition for success in progressive betting is having a significant starting capital to significantly reduce the second order bankroll-bankroll advantage the house may enjoy. (i.e., there's a very good chance you'll get felted if you start out with 3 units, vs, if you start out with 1000 units).

But alas, the casinos know that all of this takes a lot of confluence to work in the patrons favor. It's virtually impossible to isolate the odds properly and it takes an immense amount of discipline to execute the system. Throw in alcohol, breakdowns in discipline, the houses bankroll advantage and it's usually only a matter of time before they bust you out.

This type of analysis can be incorporated into strategies that feature long strings. The longer the average string, the more plentiful the longer strings, the more profitable a progressive betting strategy will be.

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Advanced mathematics can certainly be used to show that this is nonsense. You cannot make money by betting on an event that follows a binomial distribution, such as flipping a coin.

Please have a look at: Richard A. Epstein, Theory of Gambling and Statistical Logic

If a gambler risks a finite capital over a large number of plays in a game with constant single-trial probability of winning, losing, and tying, then any and all betting systems ultimately lead to the same value of mathematical expectation of gain per unit amount wagered.

Progressive betting systems can only be used, if there is a positive or negative (auto)correlation between successive bets. For Black Jack betting systems the notion of strings makes sense, as the winning probability after a losing game is slightly increased.

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Ummmm, no. This doesn't address the string nature. It's true, that when any single event is considered by itself, the probability is the same. But when strings of events are considered and approached as such, the odds of strings are substantially different than individual events.

Simply put, the odds of the next single event are 50%, but the odds of 2 stringed events together are 25% and so forth and so on. When the system approaches it from a set of events, rather than a single event....varying bet amounts has an effect on the outcome.

It's exactly why string analysis is featured in most profit analytics. A strategy that features long strings can be made to take advantage of varying bet strategies.

If a whale engages in playing roulette, he has the same negative expectancy as the sardine. When using a progressive betting system, such as a Martingale, the whale will indeed have longer positive runs (that is a higher winning percentage), but in the end the single loss that he will suffer will be much larger than the loss experienced by the sardine.

There is an analogy with trading: If your profit target is 1 tick and your stop-loss is 20 ticks, you will certainly have more winning trades than losing trades. But the average loss, you will experience offsets your advantage coming from the high winning percentage.

It is true that you need a considerable starting capital and an immense amount of discipline, to exploit any edge. However, if there is no edge (positive expectancy), discipline and capital will not help you a lot. In the long term you will lose.

The second-best strategy, if you do not have an edge (negative expectancy) is to put all your gaming capital on a single bet. If you win, just leave the casino. The best strategy is not to engage in any bets.

One last recommendation here for those who are interested: Well known Gambler and Hedge Fund Manager Edward O.Thorp has written a book on "The Mathematics of Gambling". You may download your personal (legal) copy here.

If you have a binomial distribution, this indicates that the single events are stochastically independent, or otherwise put uncorrelated. In this case there is no progressive betting strategy, which will give you an edge.

Progressive betting strategies only make sense, if subsequent events are correlated. This is not the case with a coin flip. Your example is simply false. Sorry for my lack of diplomacy.

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Agreed. I see your point now. With BJ strings, the results aren't exlcusive of one another.....one win or strings of wins indicate an increased probability of future losses.

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Agreed and that's how some of the negative progressives work. You simply keep increasing your wager to cover all previous losses until you end up winning, hoping that the house doesn't go on an infinite run. But as you said, you simply need a large bankroll to and an iron stomach to weather the swings, which mount quickly.

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I do some (gambling) counselling work with some people who have firm convictions about their ability to consistently make money from Roulette and other Casino games where they have no edge. Mostly they have devised some variable bet or Martingale type of money management approach based on capitalising on strings which they have had some winning experiences with. I work many of with these people in prison, not all. I have done quite a bit of spreadsheet simulation and research into this so as to be clear in my own conviction about it. Agree with FatTails on this one. But RM99 I am interested to see more of your mathematical explanation for your belief.

The only thing you can do is to backtest a setup, then calculate

- the average winning trade
- the average losing trade
- the winning percentage

From these you can calculate an expectancy for each trade, which describes the past.

This is the point where it gets difficult. Any approach that has been profitable in the past, will not necessarily be profitable in future. Market conditions change, technology modifies the market place, the behavior of market participants may be different.

To increase the probability that the backtested results are applicable to the future you will now conduct

- a walk forward analysis
- and a Monte Carlo simulation

This will give you a new set of expectancies, to compare with the first one. Typically these expectancies are already less favorable.

Armed with positive expecntancies from backtest, walk forward test and Monte Carlo simulation, you may now start to trade.

You trade a probability that your prior calculated probabilities still represent the actual probability. Unfortunately, no probability can be calculated in an explicit way.

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Histogram of the daily log returns of the S&P 500 for the past 10 years, as well as a QQ plot. Created using NT7, the ExportData stratgey, the LogReturns indicator, and R:

Summary Stats:
> summary(SP500)
LogReturn
Min. :-9.470e-02
1st Qu.:-5.695e-03
Median : 6.967e-04
Mean : 1.645e-05
3rd Qu.: 6.051e-03
Max. : 1.096e-01

The mean and median are positive, which is good if you are long.

But the Fat Tails can hurt

On a serious note I take issue with serial dependency for GBM, random coin tosses, etc. The coin doesn't care if it has flipped heads 20 times in a row. It's a 50/50 shot whether the next flip will be heads or tails.

Markets, however, might care if they are up 20 days in a row because they are not 100% random.

Figuring out the part that is not 100% random is what makes you money.

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Nice chart, shows the fat tails very well. Maybe I should play around a little with R. I have never used it so far.

The median is greater than the mean. This confirms that the stock market is slightly asymmetrical. There are more days with positive returns, hence the greater median, and fewer days with negative returns, but those days can bite, as showed by the lower mean.

I would expect that for most of the commodities you will find a greater mean than median, which is the opposite to what you got for the stock market.

I agree that it is very difficult to exploit the few outliers, although trend followers can catch some of them if they are sufficiently patient. However it is psychologically difficult to endure many small losses, before reaping a large reward once and then.

Fat Tails R is worth checking out. Its free, the community is growing, and there are some great packages for it. There is a learning curve, but that is OK as you don't have pay for it every time there is upgrade. Time is on your side.

Here is another NT ExportData/R graphic. In this case exactly showing how its (theoretically) possible to exploit the "Fat Tails" on the right side of the distribution. That's another way of saying "cut your losses and let your winners run" .

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If you read further than the headlines, you will find that this thread is not about game theory, but about gambling theory.

Now, as usual I disagree. Both game theory and gambling theory can be used to understand the behavior of market participants.

Game theory typically comes disguised as moral hazard. If you hear a CEO explaining: "We have to dance until the music stops playing", this is applied game theory. Repackaging mortgages and selling them as CDOs is indeed similar to a prisoner's dilemma situation and does represent a Nash equilibrium. So the big guys act along the lines of game theory, even if they don't know it. The same is true for the bonus payments, the sequence " I win, I win, I win, you (the public) lose" is another classic of game theory. Or take the beauty contest of Keynes.....

This was a small excursion from the original subject. @RM99 wanted to discuss the impact of progressive betting strategies. This belongs to money management and can indeed increase your returns, if used properly. There are several ways of progressive betting, but all of these will not yield anything if the expectancy is smaller than or equal to zero.

And obviously the big boys follow some money management rules, whatever they are.

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This was in Futures Magazine after the housing bubble burst....

Andrew Lahde, founder of Lahde Capital Management, a hedge fund that earned 870% in 2007 by shorting the type of mortgage backed toxic instruments that have left large institutions at the Fed's doorstep tin cup in hand announced his exit in October and thanked those who allowed his success: "The low hanging fruit, i.e., idiots whose parents paid for prep school, Yale, and then the Harvard MBA, were there for the taking. These people who were (often) truly not worthy of the education they received (or supposedly received) rose to the top of companies such as AIG, Bear Stearns, and Lehman Brothers and all levels of our government. All of this behavior supporting the Aristocracy only ended up making it easier for me to find people stupid enough to take the other side of my trades. God Bless America!"

I think they teach gaming theory in them math departments at them Ivy League schools...

I'm just a simple man trading a simple plan.

My daddy always said, "Every day above ground is a good day!"

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While game theory can be used to describe the crowding of the fools, its knowledge does not necessarily induce any change in Behavior.

Here is a clear conflict between social and monetary norms. The temptation to defect - term used in line with Axelrod's "The Evolution of Co-operation" is simply too large, so those guys cannot resist. Finally they did not care about the well-being of their employers, but just wanted to reap short-term bonus.

Moral Hazard is promoted by ignoring tail risks. And it is promoted by a prisoner's dilemma. This is the prisoners dilemma:

At your employer, most of your colleagues trade toxic instruments for short term bonuses. You know for sure, that in the end all the CDOs, CDSs will end up somewhere, so it is a game of musical chairs.

Option 1: You participate in the game. If it works out, everybody including you will get a bonus.

Option 2: You decide to do something reasonable, which will produce a lower yield, but has no tail risk.

Now let us look, whether you are better or worse off with option 2:

Scenario A: The toxic game continues, your employer makes a huge profit, your colleagues get a large bonusn and you get a small bonus.

Scenario B: The toxic game goes wrong, your employer loses a lot of money and nobody gets a bonus.

This shows that with option 2, doing something reasonable, you cannot win. This is actually a classical n-person prisoner's dilemma. It can only be overcome by repeating the game with the same participants many times. Social norms - which are always in conflict with unregulated markets - can by itself only be established, if the prisoner's dilemma occurs repeatedly.

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My 2 cents; let me know if I am missing the boat on this one I wont be offended. According to my wife I miss the boat a lot! lol

The one thing I don’t think these theories take into account is human perception, intuition and emotion. Many of them assume you bet on every single hand. This is not how real life works. Take the flip of the coin, if you bet on every single flip over a long period of time it would probably normalize and you end up somewhere around a 50/ 50 split. This is assuming you had zero emotions when betting. Now suppose you didn’t bet on every flip. What if you only bet when 3 heads in a row followed by a tails came up and even then you had the choice to bet or not to bet. Now throw emotion, intuition, perception and the confidence of the individual into it. This would change the odds and the outcome dramatically.

As the old saying goes you have to know when to hold them and know when to fold them. When playing roulette I only bet on black or red. Rarely if at all, do I bet on every single roll of the ball. I watch the numbers come up and look for a pattern. Its exactly the same as looking at a chart and waiting for a high probability set up. I see a cluster of numbers where I perceive to have a better chance of winning then I will bet on it. I also have confidence when I walk into a casino that I will not lose any money. Over time this has allowed me to come out ahead. The one thing about this process it can be very time consuming and some days there is very little opportunity, again this is just like trading.

I guess my point it is theory is great but how practical is it in real world applications?

SD

nosce te ipsum

You make your own opportunities in life.

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I'd agree with you totally. I am simming a scalping system(not one of my own design) and it uses small targets and larger stops.

The biggest factor in it's success is fills. Since it uses limit orders some you know for sure, would be filled, some a question mark.

Now gaming theory would suggest it's a loser, and if you took every trade all day, I have absolutely no doubt it would be, But.....

As traders we can use are trading knowledge to change that. First, there are times of day, the market is more liquid then others. Only trade then.

Second, the system will usually have 3 or 4 winners and then a loser. So take to trades and stop until there is a loser. That alone has a drastic effect on the results.

Third. Trade several markets. That way a loser can be offset.

Fourth This is the biggie. Set a very realistic profit goal for the day. I am using $75 per contract. When that is his, cancel orders, and shut it down.

Fifth. Different systems, may or may not work for an individual. Make sure the system fits you. Your belief system have a huge impact on results.

Pete

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You'd better not go into a casino. It is impossible to identify pattern with a roulette wheel, so you are going to lose money sooner or later. So your pattern recognition skills just lead you astray in this case and you make a fool. The only thing that can save you is sheer luck, as described by W.Somerset Maugham in his novel "The Facts of Life".

Seeing clusters in in numbers generated by a roulette wheel is a trap. Evolution has built our brain for pattern recognition and we even see pattern, where the are none. It is by far cheaper to count the little sheep in the sky than

Nota: There was at least one exception in history. One guy had the patience to record thousands of wheel runs for each of the roulette wheel in a casino and discovered that one of eight wheels was slightly uneven, which in turn increased the probability for the outcome of a specific number. This guy actually won a lot of money, before the casino exchanged the wheel. But you cannot make money from an even wheel in the longer run, as it is against the laws of probability.

This is an example where both psychology and money management cannot help you, as the odds are already against you, before you can make any further mistakes.

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This is not evident. You need to show first that the trades are negatively correlated. If not, you have simply become a victim to the Gambler's Fallacy.

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This would change the odds and the outcome dramatically? Only of your account, not the result of the coin toss. The coin doesn't care and has no memory.

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I agree with @MXASJ. Emotions do not have an impact on the outcome. You may even elect not to play.

Your conclusion is just an example of the Gambler's Fallacy. It is simply false (sorry I sometimes enjoy being blunt, if it serves the purpose, please forgive me).

Emotions may have an impact on correlated bets, but your example is built on uncorrelated bets. All emotions, perceptions and confidence will not change the outcome, it will be simply random.

Roulette (fair wheel) also involves successive non-correlated bets, so emotions will not have an impact on the outcome either. However, you may increase your odds, not to lose money in Roulette:

-> best decision : not to play
-> second best decision: play only once and bet everything you have on black or red (winning probability 18/37 for European Roulette, losing probability 19/37)

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Some people think that these probabilities change, once a string of TTTT has occurred. This is the Gambler's Fallacy.

The probability to get an H after a HHHH string is the same probability as the probability to get an H after a TTTT string. The coin does not have a memory, or otherwise put, the coin tosses are non-correlated or stochastically independent.

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This thread reminds me of a remark made to me by a trading mentor. I had personal knowledge of his trading success, and he used several different indicators including a form of linear regression.

At the time he was trading a 5 minute SP chart, with no over night data. So using a 10 period linear regression indicator it was obvious if there was a big gap the indicator was very wrong the first 10 bars.

I wrote him and explained how the indicator was wrong and shouldn't be used in the first 10 bars if there was a gap. I'll never forget his answer.

It was, "If I understand all you do about this indicator, I might not use it to help me make money, but I don't understand what you are talking about, so I use it, and it does help me make money." This is a real trader talking.

So to me while the discussion about the advanced math and gaming theory is interesting, to me, wanting to improve my trading, it is really irrelevant.

Pete

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If what everyone said is true then for the past 3 years I have been the theoretical and statistical anomaly that keeps statisticians up at night… And I am ok with that.

I actually have more to say, however it is pizza and movie night with the family.

And just for the record I am not giving up or giving back the 135 dollars I won 3 months ago at the Grand Victoria casino using the before mentioned technique, no matter what the theory says!

Again, no one is disputing the probability of a given singular event, we're discussing STRINGs of events. The probability of strings is a different consideration.

You can calculate the probability of a string of events. That is the concept at work here.

The law or large numbers says that the probability of a 4 series sample is just as much H-T-H-T as it is H-H-T-T,

but where the theory falls short is when you DON'T have an infinite sampling. When you have a finite sampling, you observe strings of events. It's those strings that allow a progressive strategy to be beneficial.

The idea is to take advantage of enough events that strings are relevant, but not enough that they become irrelevant (if that makes sense).

I fully concede that this strategy becomes more prolific with card games, because the prob of a win in blackjack increases with every loss...thus they are not mutually exclusive.

The theory becomes particularly relevant in examining probabilities that aren't even. A 70% win ratio for instance will yield a significant number of positive strings. If you utilize a progressive strategy along positive strings, the yield is MORE. It's simple math.

However, as I've pointed out in other threads, in order for the theory to be effective, you need to have a fairly resolute/continuous instrument. Futures contracts for example, are relatively discrete. A CL contract with it's marginal reserve will NOT allow you to "rollover" profit amounts like stocks. (i.e. you can't increase your "wager" with a futures contract like you can with stocks....because of the marginal reserve).

The other part is that the strategy must incorporate a similar profit/loss ratio. If you're strategy incorporates a 50/50 ratio, and you're anything above 50%, then string theory and progressive wagering INCREASES your rake.

For example, in a 100 trade sampling, at 70%, you'd expect the string distribution to look something like this....

Losing Series:

1 - 18
2 - 3
3 - 2

That means that you have an 18/23 (78%) probability of a winning trade after a losing trade. Losing series progressive strategies aren't as effective, because you have to have additional capital in reserve in order to take advantage (i.e. you'd have been better off simply risking your reserve over all trades).

However, as I said, with stocks, where you can rollover your profits more resolutely, winning string progressive strategies will help.

A 4 string winning series for example, will yield .25, .3125, .3906, .488 and .22 (for the last loss) for a total of 1.66 units of profit. That same series when only using the same starting positionsize will end up making 1.5 units (for 4 wins followed by a loss). It simply helps to ensure that you have increasing profits without risking ALL of your past profits on every subsequent trade. The phenomenon is even MORE pronounced when you use 100/100 ratios.

Furthermore, a progressive strategy for wins (and a reduction to the original positionsize for losers) helps to ensure that losing strings are minimized.

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For the fair coin example, strings or the number of prior samples is irrelevent as the outcome is not serially dependent on history. Its a straight forward binomial tree.

In cards, that is different. Each card drawn changes the probabilities moving forward. If an ace is drawn, there are no longer four aces in the deck.

In slots, that is different. I'm no expert on gaming regs but the slots are required by law to return x over n runs.

So the string thing only works if the future outcome in contingent on prior outcomes.

Progressive betting on the future outcomes is then a seperate issue.

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What is the probability of rolling a pair of dice 154 times continuously at a craps table, without throwing a seven?

The answer is roughly 1 in 1.56 trillion, and on May 23, Patricia Demauro, a New Jersey grandmother, beat those odds at Atlantic City's Borgata Hotel Casino and Spa. Demauro's 154-roll lucky streak, which lasted four hours and 18 minutes, broke the world records for the longest craps roll and the most successive dice rolls without "sevening out."

I'm just a simple man trading a simple plan.

My daddy always said, "Every day above ground is a good day!"

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Either events are correlated. In this case you can use strings to define conditional probabilities and execute progressive betting strategies can be used. They betting strategies will not become irrelevant with a high number of samples.

Or the events are independent, as is the case with throwing dice. In this case progressive betting is entirely useless.

This is the whole point. If the outcome of the event n (loss) has an impact on the outcome of the event (n+1), progressive betting makes sense. For black jack there is a negative correlation, as a loss increases the probability to win. Note that statistical analysis has been done on the ES, to find out, whether an up move is more likely to be followed by another up move (positive) correlation or a down move (negative correlation). The correlations that were found for ES were so small and unstable that they did not allow to compensate for transaction cost.

Again, no. The point is not, whether a probability is even or uneven, but how much that probability is affected by positive or negative auto-correlation.

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There is an even bigger problem with all this stuff, and that is the fact that the markets are moved, not by all this stuff, but by people who react to stuff with emotions.

Look at how markets react around reports, or some news crisis. On many occasions, the markets take off in one direction and then immediately come back. Then there is gunning for stops. What experienced trader isn't aware of this.

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You say there is gunning for stops. Yes, of course it is. This is the truth of trading. Now, if you are gunning for stops, all you need to find out, where these stops are located. Once they have been triggered the market is free to reverse.

I even believe that this stop triggering is part of the hygiene of a reversal. How can the market reverse from short to long, if the new longs have not be stopped out? Trading is a zero-sum game, where you simply exploit the behavior of other traders, so you need to have an idea of what they are doing and how to trap them.

Emotions are creating feedbacks. This changes the market movement from random to trending (positive correlation) or counter trending (negative correlation). Once the market is not random any more, it becomes tradeable. Progressive betting can be introduced, if the correlation is known.

Chart attached: Assumed Stops detected by indicator

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For those who believe in this 'string theory', and that it's applicable to a coin toss...

The odds of four tosses coming out "HHHH" are .5 * .5 * .5 *.5 = .0625 = 1/16

So if I flip a fair coin, and it comes up heads 3 times in a row, are you *really* ready to bet me with real money at lets say 2 to 1 odds that the next flip won't be heads? According to your theory you'd be giving me 2 to 1 odds on something which has 16 to 1 odds in your favor.

I don't think you'd be so willing to make the bet. That's because your theory doesn't work (for uncorrelated observations).

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A common tactic among trend followers (like the Turtles) is to add to winners.

The implication from this thread, if I understand correctly, is that it would not make sense to add to winners unless the fact that the instrument has started to move in one direction actually increases the odds that it will continue to move in the same direction.

If price movement is viewed as random, in other words, it wouldn't make sense to add to winners. Is that correct?

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The higher win rate is needed to compensate a lower average profit per trade

First of all, if price is viewed as random, you would not put on any position at all. The Turtles used a breakout system, so they were simply betting on the success of the breakout.

To evaluate their position they used the average true range ATR(20) over the last 20 days. When the position had moved half that average true range, they added to their position. Now if you compare the first and the second entry, the second one for sure produces a lower profit per winning trade, and a higher loss per losing trade, as the stop loss for the first lot of contracts was adjusted by half the ATR(20) as well, when they entered the position with the second lot.

So a higher winning probability is needed to compensate the lower profit per winning trade and the higher loss per losing trade! So indeed a positive correlation of time series is a prerequisite for pyramiding.

The motivation for pyramiding is not an increased expectancy but risk management

The Turtles had a stringent risk management. For the first lot of contracts they had a risk allowance equal to 2% of their account equity. The position sizing was calculated in a way that one average true range represented 1% of the account. So for the initial position they used a stop of two times the average true range.

When the price had moved in their favor by half an average true range, they trailed the stop by half an average true range as well. Doing this reduced their risk from 2% to 1.5% of their account. They would load up to 4 units, each time adjusting their stop. So after the fourth unit had been entered, the risk would be

first unit 0.5* ATR (3 times trailed by 0.5 * ATR)
second unit 1.0 *ATR (2 times trailed by 0.5 * ATR)
third unit 1.5 * ATR (1 time trailed by 0.5 * ATR)
fourth unit 2.0 * ATR

producing a total risk of 5*ATR or 5% of account equity for the aggregate position of 4 units. This gave the position a better risk profile of a unit risk of 1.25 * ATR compared to the initial risk of 2 * ATR.

So pyramiding allows for trading size, but can only be done if both conditions are fulfilled:

-> there is positive expectancy
-> there is a positive correlation between the consecutive entry levels of the trade

The Gamblers Fallacy

The Turtle approach is actual the opposite of the Gambler's Fallacy. Our gambler assumes that after three consecutive losses there is an increased probability for a winning trade. If prices are non-correlated this is false, if they are positively correlated - as they typically are after a successful breakout - it is false as well.

The Turtles assumed a positive correlation for the subsequent entry levels, which was true at the time they traded their system. Since then, numerous other trading systems - including the 2B and Turtle Soup setups - have exploited the breakout traders, so today this system can no longer be traded. That explains that the rules have been published. It is a dinosaur.

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The whole concept behind technical trading is that you're attempting to quantify a dynamic system. When you use indicators, what you're essentially trying to do is put a metric on news and secondary event driven market trends.

Thus, you can and many people do model the markets to predict certain outcomes....with lagging indicators.

Case in point, it's very easy to take something simple like MACD and adjust the slow/fast lengths and then do a historical analysis of how the instrument behaves upon a simple MACD cross. You can adjust the chart time period and any other number of parameters and go back to whatever degree of certainty you like (2 days, 2 months, 2 years, 2 decades, etc).

By doing that, it's very easy to craft a strategy that proclaims 95% of the time, when the MACD crosses on this particular chart, with these particular inputs, the market will move in that direction at least x number of ticks before it retraces y number of ticks. As such, it's not difficult to craft strategies that win 95% of the time (with large loss limits and small profit limits).

Thus, you are simply using a technical analysis to give you an "indication" of the physical, real sentiment in the marketplace.

It's no different than the same intangible trend rules that many investors use like "sell in May and walk away." Cyclical analysis and the trends realized are nothing more than a general obeservation of real world supply and demand and external drivers (it's very easy to see cyclical trends in commodities depending on their relative usage during certain times of the year, etc).

This analysis can be done in virtually infinite number of means and methods (hence this website) and it's what every technical trader is trying to achieve.....a numerical representation of trend, trend strength, reversal points, etc, etc, etc.

From that analysis, you can produce strategies that have a "predicted" profit ratio/win rate, (attached with an appropriate confidence interval).....i.e. you're 95% strategy might have a confidence interval of +/- 15%, so you know that the absolute WORST case, during the WORST period tested will yield the bottom side of the confidence interval....you can then add whatever degree of safety you like to sooth your risk tolerance for future market conditions that may be undesirable. Once you've done all that, you've applied an appropriate amount of slippage and commission, and it's still profitable and acceptable to your drawdown/risk tolerance, then away you go.

So your point is taken, that markets do change, but at the same time, there's precedence and consistency, it all depends on the timelength of the analysis and what you're choosing to observe. What does the market do when it gaps up on an opening after a long weekend? Candlestick analysis....whatever...it's all an attempt to quantify the physical market sentiment on the trading floor and among consumers.

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You're still not grasping my point. I'm not interested in single events....I'm interested in the inherent strings present in samples that are not evenly matched.

It is MATHEMATICALLY impossible to observe a sample of any size that has an uneven outcome probability and still maintain the same frequency and distribution of repetitive strings. You can try if you'd like, but it's impossible. You can't have an event that's 90% outcome and 10% opposite outcome (binomial) and observe the same number of strings on both sides....something has to give, you either didn't observe 90% or there are more frequent and more depth to the strings on one side.

THAT is the concept at play here, NOT the probability of a single event. Where many mathematicians get lost is either in trying to reduce it to a singlular sample size or increasing it to an infinite sample size, neither of which is useful or feasible.

Like you said, string progressive strategies are mainly for risk management....under MOST cases (as I outlined above) you'd be better off simply leveraging as much as you can for a winning strategy (> 50%), howver, when the profit/loss ratio is comparable and when the profit/loss amounts are large, the phenomenon DOES become useful for increasing profitibility.

If you have a 70% win rate and you know you're going to encounter more strings of winners than losers and the strings will be larger, then a 50% "rathole" method ensures not only that you make more profit on winning strings, but if you reduce your positionsize to the original starting amount when you lose, it also helps to minimize losses on losing strings.

Obviously, the greater the disparity in the win/loss rate, the less the losing strings play a role. (on a 90% win rate, you end up with a 1% chance of having 2 consecutive losing trades over the next 2 trades...that's not to say it won't happen, it's just to give you an indication of how unlikely it is....)

As someone else pointed out, this phenomenon is particularly useful in card games that are NOT mutually exclusive, where the outcome tends to happen in strings (because the more wins in a row, has a bearing on the outcome of every subsequent event). Progressive strategies (50% winner, return to 1 loser) try to take advantage of winning strings and minimize losing strings. Where THAT type of strategy can actually be detrimental is if there is a perfectly even distribution of winners and losers.....and this is where the mathematicians get confused.

The law of large numbers says that over time, the probability that you will see "W-W-W-W-L-L-L-L" is the same as "W-L-W-L-W-L-W-L" and that is correct. HOWEVER, when you're not dealing with infinite sample sets, it becomes VERY likely that you will observe an uneven distrubution....the higher you go in sample size, the more accurate you will see the predicted win outcome (with flipping a coin, the first probability with a sample size of 1 is either 100% or 0% and then begins to approach 50% from then on....assuming a perfectly random, perfectly binomial coin flip outcome). The trick is to increase the sample size so that you can get close to the predicted win rate, but not so much that you negate the tendency of uneven distribution. THAT is the secret in progressive strategies.

A sample size of 100 coin flips might yield 51 heads and 49 tales, but there is a HIGH degree of probability that you will observe strings of wins and losses over that sample (and a very low probability that every loss will be followed by a win and vice versa).

As the win/loss ratio increases, this phenomenon becomes considerable. As I stated above, you cannot observe a 70% outcome rate for a binomial system and have an equal size and distribution of string outcomes....it's physically impossible.....so instead of saying that it's highly unlikely that each win will be followed by a loss, it becomes impossible (in retrospect) which translates into...."if your strategy truly does achieve a 70% win rate, there's GOING to be more strings of winners than losers."

I can't explain it any better than that.

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Progressive betting can be applied, whenever an event is contingent on the preceding event. You cannot apply progressive betting to any coin flip experiment, because there is no contingency.

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Thanks for the interesting post. This was a new idea for me and I wasn't really sure I understood the implications so I created a quick spreadsheet to let me experiment. I've attached the spreadsheet if anyone is interested. Changing the sequence of wins and losses in column D will show the impact of different strings on trading capital.

I can see that the net effect of this kind of a system is that the profit from a string of winners is greater than the loss from an equivalent string of losses.

It's not clear to me, however, whether this effect is significant enough to offset the losses that occur during periods of "chop" when the results oscillate from W to L without producing any strings. Each "W-L" sequence results in a net loss of 5 units. I don't have the mathematical skills to prove whether the amount that you "bleed" during non-stringy periods is expected to be more or less than the amount you gain during strings.

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Fat Tails,

This is a great argument you are making and I think it applies to almost any system out there once they are revealed. There are actually scalpers and trading desks (or even program tradings) out there that, simply put, trade in the opposite direction of the most popular retail systems' signals for a few ticks sure bit. I recall there were studies on profitability of some popular indicators like MACD crossover, for example, that showed trades taken in the opposite direction of the signals generally performed better.

I think your argument explains why someone with a working system would even consider selling it to anyone. You know, miners used to kill claim jumpers here in USA. Everyone out there selling systems and indicators are focusing their promotion on money management and education (and there are many of them) and taking advantage of the trading crowd psychology that huge money can be made easy. They know their tools and services don't work. Otherwise, with their own system and money management techniques they will be vacationing on sandy white beaches and trading more contracts for more money.

The question is: what is the edge and how you find it since all trading systems work only for some time? I think one should look into the seasonality edge that is based on cyclical natural phenomenon (mostly weather) and basic consumer commodities, i.e. wheat, corn, soybean, etc. The farmer bets the farm (no money management), plants beans, and hopes for a good crop at the summer's end. He hedges himself with buying some put options and waits. This seasonality edge is available to the farmer and anyone with patience trading for a living. We all want the quick bucks right now, and that is not going to happen with a system or without any luck. There are many references on seasonal trades that show favorable outcome trading seasonality. Just increase the size of your bet if you hear in April that Mid-West is too wet or there is no rain in sight for planting soybean. Just an idea for a seasonality system! No one can fool or manipulate the mother nature. We need to use it as the edge. Did anyone short the uranium stocks three weeks ago on Japan's disaster? Yes, it is cruel but an edge.

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TMFT,

Interesting! But, that will not happen in Vegas. The roller and the dice pair are changed if the pit boss detects a streak beyond the normal. Is that the same as the trading desks on the other side watching your moves?

The animals and plants that populate the Earth now, are not the same species that were living here some hundreds of millions of years ago. Man has found a - temporary - edge to exploit other species and cause a lot of havoc. The game will be over, when

- either a new species will feed on our success
- or our own success has caused the species that we feed on to disappear

Many strategies fed on the successful breakout traders of the 80s and 90s, and in the end simple breakout strategies became difficult to trade. It is all about evolution.

How could you identify when these strings would be most probable to occur given that there is a fixed 50% in each flip? Seems like simple math so I'll be blown away if you have an edge.

Six years ago I was on pace to become well off from online poker but my game couldn't adapt to make it worth it anymore. It was hard to throw in the towel.

ESPN's WSOP program got people interested because they gave away player's hands and only showed exciting pots. I rode a 2 year poker bubble of easy money at pretty high stakes driven by loose credit, covenant access and lack of government regulation. Now that the pots are 1/3 the size and people are betting small ball - forget about it.

No. Again, you're looking at the wrong resolution. I'm not making ANY predictions about singular events. I'm talking about STRINGS.

For strategies that have shown/observed a greater than 50% win rate, it's mathematically impossible to have an equal number or depth of strings for winners and losers. Thus, you know that going forward, IF you are able to observe a greater than 50% win rate, then there will be more positive strings and they will generally be of greater length. The higher the win rate, the more pronounced the phenomenon becomes.

Your argument is exactly why the progressive strategy works. I do not have a considerable edge in determining the next coil flip (50/50), but I do know that if I progressively increase my bets 50% on each winner, over the course of strings, I can take advantage winning strings and minimize losing strings.

it's like this.....

What if I bet you, 10 to 1 odds, (My $1000 against your $100) that if you flipped a coin 10 times, that the distribution would NOT be even (i.e. that you'd observe H-T-H-T-H-T-H-T-H-T exactly or vice versa with T starting)...would you take it? You'd be a fool if you did. The odds of that happening are .5^9 or just under 2%. That would mean that I was giving you 10:1 on a 50:1 probability, a good bet for me.

Heck, even a shorter string, it becomes apparent. what are the odds of tossing 4 times, and having the outcome be EXACTLY H-T-H-T or T-H-T-H?

.5^3 or .125 or 12.5% or 1 out of 8 times roughly.

That is to say, that 1 out of 8 times, the prob will be exactly even and evenly distributed along the sequence.

With a coin flip, over the course of infinite sample series, the even distrubution negates the string contributions from the progressive strategy (i.e. for every time there's a string that helps a 50% add to winner strategy, there's another string of oscillating outcomes that has a negative effect for the same strategy).

However, as I said, with outcomes that are NOT 50/50, this type of strategy starts to have merit.

No, the probability of any individual outcome is still 50%. So you cannot predict anything going forward. However, you know that the likelihood of an uneven distribution (for a finite sample set) is MORE than the likelihood for an even one....so you know that if you stick to a string progressive strategy, it works out.

The problem here is that the longer you go, the more an oscillating outcome will hurt you (for every string of 3, a H-T-H-T sequence hurts you)...in cards, as discussed previously, there are more inherent strings becaust it's not mutually exclusive, a winning hand in BJ indicates an increased probability in losing hand (which is why/how people tend to count cards).

However, as I stated earlier, for outcomes that are NOT 50%, the strings HAVE to be unevenly distributed, regardless of the sample size (if you truly observe win ratio above 50%).

I agree with this. I've walked through the logic and convinced myself that if you have say a 60% success rate and therefore can expect more winning strings than losing strings over a large sample, this progressive betting method could make good sense.

The hard part is determining in real time, amid changing market conditions, what your system's success rate is right now. You can develop pretty good idea of the system's success rate in the past, but you're up against a constantly moving target. A drawdown could be an unlucky streak, or alternatively the first sign that the market has changed in a way that removes whatever edge you would have expected based on the testing process.

That's just my own psychological bugaboo though and not a commentary on this progressive betting system in general.

Dangerous conclusion. A 60% success rate does in no way justify on its own to make use of progressive betting.

Take a dice and have a look at the two events

-> 1 or 2 or 3 or 4 -> you win (probability 2/3)
-> 5 or 6 -> you lose (probability 1/3)

Now your winning probability is about 66.67%, but there is no way you can make use of progressive betting, as the next event (throwing the dice) is not contingent on past results.

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Again, no one is disputing that. What I'm simply saying is that IF you develop a strategy based on experience, backtesting, whatever, and it enjoys a greater than 50% win rate in the past, and you have a reasonable expectation that it will in the future...then strings become useful.

What you're essentially trying to argue is that there's no way of predicting the outcome of a single event, and that's true....but I'm not talking about singular events...I'm talking about sequences of multiple samplings and the observations you gain.

Furthermore, just because you enjoy a 60% win rate over a given set of trades...everyone knows that it's not a guarantee you'll see that in the future, but it has an "expectation" which is exactly why we all trade....if it was ALL truly random and unpredictable, then we'd all be fools to try to make money at trading (or we'd simply look at someone who makes money and chalk it up to blind luck).

If you have an observed edge, then string theory becomes useful. It does NOT however always result in an increase in profitability, you have to run the numbers to see....

Like I said, for a 95% win rate for a strategy that makes a small amount 19/20 times but then loses a large amount (on average), by progressively increasing your positionsize by 50% of your previous winning, you would in fact make less money than you would if you were to simply increase your positionsize each time by the full amount of profit realized on the previous trade.

Progressive strategies only increase the profitability of systems dealing with events that are not mutually exclusive.

However, what it WILL do for you is decrease risk and drawdown. A 95% strategy that goes say 5 profit/50 loss, (assuming it's still able to overcome comissions and slippage) will be able to reduce some of the drawdown experienced by increasing positionsize only 50% after each successful trade, rather than the entire 100% of profits....

Additionally, returning to the "base" or first positionsize for a loser, becomes a very smart risk management strategy for minimizing losing strings (more so for strategies that are closer to 50%, because with a 95% winner, the odds of consecutive losing strings are much less).

When you evaluate these methods based off account size, rather than positionsize, you will see that it does not increase your profitability with respect to account size (i.e. if you had enough in reserve to increase your profit amount after a loser, you'd have been better off simply putting that reserve in play for all the positions)...but again, it does help to "smooth" things out a bit. By "rat holing" 50% of your profits each time, you "ensure" that you're retaining at least SOME profit.

This becomes VERY useful when the profit/loss goals are very large with respect to account size.

Let's say for a second that your broker allows you to trade 4 CL Contracts with a $5k account (yes they are out there), that means that every 125 ticks, you earned a 100% margin (minus fees and slippage).

So if you're strategy had a profit goal of 125 ticks, a loss limit of 125 ticks and an "observed" historical edge of 60%, then what you would discover is that there were strings of events in your favor. IF that edge were to continue, and over the course of the next n sequence of trades you were to up your positionsize by 50% after each win, (and return to the original positionsize for a loser) you'd discover that it made more money.....as increasing your position size by all the profits would eventually result in you losing ALL of it back once a loss ocurred. Furthermore, a string of losses would be catastrophic. By varying positionsize, you CAN take adavantage of strings (not singular events, but STRINGS) to smooth out your equity curve and ensure you make sustainable and growing profits, rather than grow like crazy, only to see it all burn away very quickly.

I think it all hinges on whether there is any real relationship between subsequent trades, where the outcome of one trade can affect the following trades. If the trades are all independant of each other then Fat Tails is completely correct, there is no edge to be had from progressive betting or martingaling. Doesn't matter if you have a winning system or not; if there is nothing in the system that causes a relationship between trades, then there is no advantage to be had doubling down or progressive betting.

However there may be situations where there is a real relationship between the results of a trade and the results of subsequent trades. As someone pointed out, Blackjack is a game where there can be a real relationship between 'trades' because the game uses a deck of cards which gets depleted and therefore the odds change as the composition of the cards remaining in the deck changes.

In trading, especially systems that are in the market most of the time and take every single setup presented without skipping many, there can be situations where the results of one trade do affect the probabilities of subsequent trades. For example if a trend following system that takes every trade suffers a string of losses but continues to take every single trade, then the probability that each new trade will be a winner will increase somewhat. This is one big reason it is so important for these types of systems to take every single trade and not cherry pick only some of the trades.

Same with a mean reversion system that keeps taking the same trade even after a loss. Eventually , if you keep trying over and over on the same MR trade, you will finally catch that falling dagger and your odds should improve with subsequent attempts.

David Viraldi at CSS Analytics recently posted an interesting indicator idea that relates to measuring this tendency for trend following systems.

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No. This is the fallacy. It only can become useful if the outcome of the current trade depends on the outcome of the prior trade. In case there is no dependency (example coins, dice, roulette) it is not useful. Progressive betting is only useful, if the trades are (positively or negatively) correlated, and if these correlations are stable.

This is exactly the point.

It is easiest to understand the difference by comparing dice and black jack.

Dice: The current event is not contingent on the prior event. Even if you had a long string of losing bets, this does not increase the probability that the current bet is a winning bet. Progressive betting cannot be applied. The idea of strings is a fallacy.

Black Jack: The current event is indeed contingent on the prior event. Cards once played cannot be played again. This is why a string of losses changes the probability that the current bet is a winning bet. It makes sense to count the cards (by attributing them a value) and use progressive betting strategies. Edward O.Thorp has described this in his book "Beat The Dealer".

It is not a question of a high or low winning probability, but a question whether consecutive bets are correlated or not.

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Even if the events are mutually exclusive it follows.

here's an example.

Let's say you had a typical 6 sided die....and if you rolled a 1,2,3 or 4, that's considered a win. If you rolled a 5 or a 6, that's considered a loss. The predicted outcome would be 66% yes?

So if you were to plot a series of 100 rolls, you'd find that there were not only more wins than losses, you'd also find that the strings of wins and losses were different as well, there'd be MORE winning strings and the average length of a winning string would also be more.

So, if you know that the string of 4 wins, is much more likely than a string of 4 losers, now you can start to realize the phenomenon of progressive strategies.

If you were to increase your "bet" (positionsize) 50% for every winning trade, you'd realize MORE profits than if you were to simply wager 1 unit. Moreover, if you were to return to the original 1 unit wager each time you lost, you'd MINIMIZE the losing strings and only lose 1 unit for each loss in a successive losing string.

You have no future expectation of any single event...they are mutually exclusive, the probability of a winner on the next roll is the same as the roll before. Again, you're looking at it from a single event. I'm not approaching the strategy/progressive betting system from a single event. I'm approaching ALL events with the same strategy.

Your point would be taken, if I were to try to "cherry pick" and increase bet size once a certain number of wins or losses had been achieved. THAT would not work, as the prob is still the same for the next roll.

However, if I employ a progressively increasing bet for wins, and a return to 1 for losers, over time, it WILL yield more than simply wagering 1 unit over all rolls.

The problem here is again, in order for this to work, you have to have an equal profit/loss ratio and you either have to have reserve on hand to increase the wager sizes or the profit/loss amounts have to be significant to the size of the account.

Thus, the strategy WILL NOT increase you're overall profit margin (if you needed additional cash on hand to employ it, you'll find you'll make MORE profit by simply wagering as much as you can on every bet). However, you'll find that employing that strategy helps keep you from getting "felted" (your chips taken to zero) in the event of a large run of losing outcomes and also help you to ensure you keep at least some profits on fortunate winning strings.

It's simply a money/risk management strategy.

I cannot explain it any further than that.

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Betting 2, or even 200 units per roll (assuming you have a system with positive expectancy) will similarly yield more than simply wagering 1 unit over all rolls. This does not mean you improved your odds. All you did was increase your position size.

Correct. We need to differentiate between progressive betting and position sizing:

Progressive Betting

The bet size depends on the outcome of the prior bet or the outcome of strings of prior bets.

Position Sizing

The bet size depends on expectancy and win rate. The bet size is regularly adjusted to match account equity (see fixed fractional position sizing as suggested by Ralph Vince or Kelly criterion).

The Difference

Position Sizing is a money management technique, progressive betting can be used to increase the expectancy of a bet, but only if the bets are correlated. The expectancy can now be calculated as a conditional probability as opposed to an absolute probability.

I ran some numbers through my spreadsheet and gave this a bit more thought. I find myself agreeing more with Fat Tails now.

If your system has a winning percentage greater than 50%, then the progressive betting technique will improve profitability. However, this isn't due to the concept of capitalizing on "winning streaks" but rather, more accurately, to the simple fact that under the progressive system, you're using an increased bet size at least some portion of the time. If it's a winning system (which we're assuming by definition here), then larger bets equals larger expected profits. Presumably a similar improvement could be achieved by increasing the bet size at random intervals, regardless of the outcome of the preceding bets. And of course profitability could be improved more by increasing the bet size on all trades (assuming you can afford to do so).

When trading in the real world rather than a thought exercise, however, there does still seem to be some value to the notion of varying bet size based on recent experience, particularly when it comes to decreasing bets during a drawdown. That's because you don't really know that your system will maintain the same success rate you've observed in the past. If recent results are giving you evidence that your method isn't working at the moment, it still may be wise to scale back until proven otherwise. Or maybe this is just "wimping out."

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All I'll close with is saying that it's a similar concept as performance enhancing drugs in baseball.

There are idiots out there that say "taking steroids won't help you to hit a baseball." and the intuitive logic tells you that if it didn't help, they wouldn't do it. Steroids helps to increase production in a number of ways, namely extending careers (which increases overall career numbers) allowing a player to recover faster from injury (thus improving seasonal and career numbers) and it also helps to improve a players average distance on their hits and also allows them to put on additional muscle mass through more intense workouts, which in turn, helps them to use a heavier bat with the same swing speed or increase their swing speed using the same weight bat. In just about any aspect, it DOES help with homerun production, but there's always an idiot who goes "taking steroids won't help you to hit homeruns."

If it didn't help, then they wouldn't do it. You don't see baseball players all on a vegetarian diet....you know why? Cause there's no evidence that it helps them perform. If it did, then they'd do it.

The same concept is true here, why do you think virtually every performance report gives string and winning/losing sequence stats? Because it is an effective way to reduce drawdown and manage risk.

If you don't believe me, try it yourself. Take your own winning strategy, apply a 50% positionsize increase (or take 50% of your previous profits) following winning trades and then return to the original positionsize following a loser. Plot the equity curves and compare. It's not that difficult.

I read thread this yesterday and found it very interesting. I used to have discussions like his with a programmer buddy who was a Math PHD. He hated TA and insisted markets were random so no prediction could work then got sick of modelling passenger queues for Airlines and decided to make a living selling options. My argument was always along the lines that TA is not about prediction but that even random events have trends and all you are doing is finding and riding them.

Anyhow I digress. I happen to believe in changing R size in accordance with wins in row even though up until now I have never bothered to prove or disprove the theory.

It seems to make sense that it should work given that even at a 50% win rate you can expect a string of 14 wins in a row at some stage. The down side is that the 1st loser after that is a doozy.

Anyhow I did a little Monte Carlo to see what would happen. Starting at $50,000 and risking 1% of capital I increased R by 0.1% every time there was a win and went back to 1% as soon as a loss came along. I did this for 1,000 random trades and then took the results for 1,000 of these and stuck them into a chart.

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Just in case you were wanting to give up on this method of position sizing. Have a look at what happens if your win rate is 55% - Hint. Look at the scale along the bottom

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If you do this with a win rate of 50%, increasing and decreasing R cancel out in the longer run.

If your win rate is 55% always assuming that the average profit equals the average loss, this does not cancel out, as the results are biased to the upside. So you are gradually increasing the bet size knowing that your bet is favorable.

Actually you get 11 wins for 9 losses, so after 20 runs you will typically have increased your bet size from 1% to 1.2%. After hundred runs your betsize should be close to 2%, after 1000 runs close to 11% of the original equity.

This is similar to fixed-fraction position sizing, as you add to your bet for every winner. It is part of the anti-Martingale strategies an can be applied to uncorrelated bets. It has nothing to do with strings though, as strings rely on correlation between consecutive bets.

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You're still not getting it. Call it semantics, chicken or the egg, whatever...but you cannot observe an outcome rate different than 50% and not observe an uneven distribution of both wins/losses and the strings of those outcomes.

Whether you attribute the phenomenon to the rate or to the strings, the phenomenon stands, that if you approach the wager amount consistently and methodically ACCORDING TO STRINGS, it has an effect on the overall yield of the strategy.

As I stated earlier, the real world yield isn't always optimal though. In order for the phenomenon to be optimal, the wager sizes have to be fairly equal and the wager has to be large with respect to the account size (or the instrument has to have very continuous compounding).

When you try to evaluate this type of strategy modification with futures, you'll find that in many instances, you'd have been better off just wagering the maximum amount every time. (i.e. if your marginal reserve is $5k, then holding that extra $5k in reserve and putting it into play only after a certain outcome increases the yield, but not with respect to overall account size).

It works best for instruments like stock where you can compound/rollover much smaller amounts from a previous win.

It also works best when dealing with win rates that are larger (but not too much) than 50%. I.e. a 90% win rate would be better served to just rollover as much as you can, because the strings are much longer. (and the losers much less common).

It's simply A TOOL (singular) that you can try and see how it effects your particular strategy. It doesn't always result in a desired effect....but in some situations it can help to reduce risk/drawdown and/or increase profits.

Again, you keep fixating on this concept of expectation and I fully conceded that for the most part, trades are independent and mutually exclusive.

HOWEVER, if you have a "side heavy" strategy that does well in bull or bear markets (and not as well in sideways markets) then a win may have some indication as to the expectation of the next win....(i.e. if you're winning 3 in a row, you can do the analysis to see what the odds are on the very next trade following a 3 win series).

Regardless, if you approach the strategy as groups of strings (rather than individual outcomes) it can and does have an effect by consistently and deliberately altering your wage amount.

@RM99 Your statement is not clear and confuses different subjects.

You really would need to differentiate between uncorrelated and correlated bets.

Strings have no meaning, if consecutive bets are uncorrelated, such as for a coin tosses.

If you apply any progressive betting system to uncorrelated bets, it just becomes a tool for position sizing.

For uncorrelated bets the progressive betting produces a similar result as fixed fractional position sizing, as you increase your bet size, when your equity has grown. However, this has nothing to do with strings, but simply follows from the positive expectancy.

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Well, the posts above tend to prove otherwise. Unless you're trying to argue that trades aren't mutually exclusive, which is debateable, but difficult to prove.

I would argue that for an average expectation of 55%, each trade has an expectation of 55%, regardless of previous outcome. SOME strategies, might lend expectation, but for a totally random outcome distribution (as posted above) and 55%, you're STILL GOING TO OBSERVE STRINGS.

This is the part that you're refusing to realize. you CANNOT observe a 55% win rate and NOT observe random strings of winners that are longer and more common than equivalent losing string occurrences.

Whether the strings make up the winning percentage, or the winning percentage is what makes up the strings, it's semantics.

The FACT IS, that adjusting position size, even with mutually exclusive outcomes, and an observed rate <> 50%, it IS going to have an effect on the yield.

Just because I have an 85% win rate, doesn't mean that after a win I have any different expectation than the trade before. THAT is approaching the decision from a single event. However, IF you approach the decision from a total string point of view, over time, adjusting position size can and does have an effect on the outcome.

At this point, if you don't agree, then we'll simply have to agree to disagree.

We agree that you always will observe strings. We agree that if the win rate is higher there will be more and longer strings with a positive outcome. But now comes the catch:

(1) Non correlated bets (coin tosses, throwing dice, playing roulette, described by a binomial distribution)

The strings are random and have no meaning. You may increase your wager after each positive outcome. You may decrease it after each negative outcome. If the win rate is > 50% you will have more positive than negative outcomes by definition. So you will gradually increase your bets, which in turn has an impact on the yield. You have just increased your position size. You will get the same result, if you make a small increase in position size with every bet, such as you would obtain with fixed fractional position sizing.

(2a) Positive Correlation: A positive outcome of the prior bet increases the expectancy for the current bet.
(2b) Negative Correlation: A positive outcome of the prior bet decreases the expectancy for the current bet.

For Black Jack consecutive bets are negatively correlated. If you have had a number of losses, this increases your expectancy which can become larger than 50%. That is why some Black Jack players are watching for negative strings by counting cards. This gives them an edge. Casinos will respond by increase the card stacks to make it more difficult to overcome the edge of the casino.

To use this knowledge for trading, you have to find out whether consecutive price moves are positively or negatively correlated. This is not easy as correlations change. In a trending market, you may assume positive correlation, in a range bound market negative correlation. This really depends on the other players (traders) in the market. There are positive and negative feedback traders, and you do not know which group will dominate during the period of your next bet.

One way to look at this are the COT reports. Commercials shows anti-cyclical behavior (with a high crude price you will find a considerable net short position), Managed Money will have a net long position. However managed money traders can also generate negative feedback, when they start rushing to the exits.

What I want to show is that the concept of strings only makes sense, when the bets are correlated. For non-correlated bets there are strings as well, but they have no impact on the future outcome of bets. So increasing bet size after a positive string is just a money management tool.

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It would be interesting to see the results of a random distribution of 55% outcome probability with a fractional position increase of 50%,

Then compare that to the same distbrution with a fractional increase of 50% following winning outcomes, followed by a return to the original amount following a losing outcome.

A third iteration would be to incorporate both, by increasing fractionally following a winner, return to original following a loser and then a "reset" or increase of the base/original positionsize once the account has increased past a certain "reserve" or drawdown amount. (i.e. return to original position following a loser, up until the account has doubled, then double the base/original position size and continue from there).

Lets try it like this. 1500 trades. Standard win/loss is 500. When there is a win then increase by 50%. Keep doing this until there is a loss then go back to 500.

I have added red blobs to show what would happen if you didn't alter the R. These are calculated as

(win% * 500 * 1500) - (lose% * 500 * 1500)

At low win rates you can make your results much worse but at modestly high win rates your results are improved by altering your R to take advantage of the fact that strings occur

This makes sense. If you had a 50% win rate and you got wlwlwlwlwlwlwlwl you would keep losing 750 and winning 500 and go broke very quickly. At 50% win rates the strings tend to be interupted by streaks of wlwlwlw which undo all the good work from a string of 5 or 6 wins in a row very quickly. the higher you win rate the less that happens so once you get in the 55% and over the altered position sizing improves your bank even though you wil get some doozies of losses. There were plenty of runs where the maximum loss is 4,000 (8 times your standard R yikes!) but the final profit is still way higher than using a flat 500

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Changing position size like this is NOT equivalent to fixed fractional position sizing. The bet size is reduced dramatically as soon as a loss comes along. With FF the be size decreases a little each time a loss comes along

i.e. 100,000 risking 1% your next R is 1,000 if you just had a loss but may be 3,000 if you have had a string of wins.

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For someone who knows excel it would be very easy to put this to rest with some spreadsheet modelling...wouldnt it?
Use the random number generator to model the win %
Run tests for w% of 45, 50, 52, 55, 60
start with $100000
initial bet size $1000
After a win increase position size by 1%
After a loss decrease it by 1%
Plot in format X axis is trade number, Y axis is equity in %
Run a simulation of 1000 trades for each w% - then compare to the curves for constant bet size
There may be more aggressive bet sizing formulae that could also be modelled.

Unfortunately it would take me days to work out how to do this..I would like to see the results...anyone?

The other way for RM99 to demonstrate application of his beliefs would be to run a forward test in a journal?

Unfortunately, like I said, it's not really feasible for instruments like CL because of the discrete nature. Unless you're trading a lot of contracts, a 10% increase in position size isn't feasible.

I'm away from home, but I'll work on an excel analysis when I get back.

I didn't think this tactic would make sense - particularly when you set yourself up for a big loser at the end of a string - but am inclined to trust Nickemp's spreadsheets. Well done RM99.

Hopefully the debate will be as rigorous as to the best possible applications for trading.

I'm wondering if adding contracts while day trading futures - but only in the direction of the longer timeframe trend - might be a better alternative than a fixed fractional position size.

What would be the best stop loss and what time should these trades be initiated?

Sorry for having more questions than answers - I just swing trade futures and stocks now.

Here is a Fixed Fractional Position Sizing Calculator if helpful to anyone.

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- if your expectancy is greater than 50%
- and if you gradually increase your position size,

this will improve your profits, compared to maintaining a fixed position sizing.

Not really unexpected.

This has nothing to do with strings, because strings are irrelevant, as long as your bets are not correlated. Nevertheless, increasing the position size, does increase returns, if the odds are in your favor. It also increase the risk of ruin, if your bet size grows faster than your equity.

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I have shown that for a given percentage win you gain is greater if you increase your position size when you are having a string of wins than if you keep your position size the same. That's why all the blue dots are on the right hand side of the red blobs exept for win/loss ratios close to 50%

The strings are highly relevant nt because the bets are correlated but because they occur all the time in random sequences. You can calculate what size strings to expect for a given win/loss rate and they are much longer than most people expect.

Your point about risk of ruin is mathmaticaly impossible. If your normal risk is $500 and you increase by $250 per win when you are betting $5000 you have just made profits of $47250 because you must have had 17 wins in a row to be betting $5000

What you have created is a bet size modulator. You let the betsize grow and then reduce it back to the starting value. This just increases the average bet size. If you have a higher win rate, it will give you a higher average bet size than for a smaller win rate.

The outcome of your bets are not influenced by or correlated to the bet size. If you have doubled your bet size this does not change the win rate of the following bet, so it just doubles your expectancy.

The bet size modulator is a funny machine, which modulates the bet size all the time for no reason. Very entertaining, but not useful for non-correlated bets. Instead of doing this you can calculate the probability for each winning string and once adjust the bet size. Let us look at an example and take a win rate of 60% and look at the probabilities for strings of wins

loss 40% -> bet size 1.0
1 win (actually a string loss/win) 24% -> bet size 1.5
2 wins (loss/win/win) 14.4% -> bet size 2.0
3 wins (loss/win/win/win) 8.64% -> bet size 2.5
4 wins (loss/win/win/win/win) 5.18% -> bet size 3.0
5 wins (loss/win/win/win/win/win) 3.11% -> bet size 3.5
6 wins (loss/win/win/win/win/win/win) 1.87% -> bet size 4.0

I have added the prior loss to make the events mutually exclusive, so that the probabilities will add up to 100%.
Your betting modulator will then vary the bet size according to the probabilities of strings. The average bet size will be.

This will probably converge to a value close to the double of the original bet size. So instead of using the betsize modulator, you can also simply double up your bet size for all bets and will get a similar result.

Where this approach makes sense

In trading the events are non-random, and there are correlations. Otherwise technical analysis would be useless junk and markets could not be traded by technical strategies.

The approach that you presented is part of the Anti-Martingale betting systems. There are two advantages linked to this:

(1) Typically the win rate of any strategy is not constant over time, but will vary. Let us assume that the win rate of a breakout strategy with defined target and stop loss increases from 55% to 70%. In this case the anti-Martingale betting system will increase the bets automatically. Then decrease the bets again, when after a few months the system returns lower win rates again. The anti-Martingale strategy therefore lets you trade size, when the win rate (in the recent past) has been higher, but reduces size, when the win rate gets lower. This is the opposite of doubling, when in a losing position.

(2) In case that you have positively correlated bets, your expectancy after a win will be higher than after a loss, so you should bet more after a win and less after a loss.

However, if bets are negatively correlated, the anti-Martingale system may wreak havoc, as it bets high, when the expectancy is low.

Think about it. It is like a puzzle.

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This is where the rubber meets the road I think. In a thought experiment you can establish the win rate by definition; in the real world the win rate can be guessed at but is subject to change without notice.

What is the implication for real-world trading? Should bet size be varied based on prior outcomes? This might be a good idea if you theorize, for instance, that the market goes through cycles when your trading method is favored and those when it is not.

Then again, we've also seen that this "string" approach can be affirmatively harmful if the win rate is right around 50%, as you get chopped up by all the W-L-W-L sequences. So if you regard the current win rate as an unknown then this approach might be a bad idea.

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Just to be clear: The issue is not the win rate, but the expectancy. You can have a high positive expectancy with a win rate of 30%, such as is typically the case for trend following strategies. If you refer to a win rate of 50% with equal outcomes (which was the original assumption), then you should not bet anyhow, as there is no edge. In this case, the Anti-Martingale approach is harmful, because it simply amplifies the outcome of "no edge".

Where it is very harmful, are bets that are negatively correlated. Imagine that you play Black Jack and you increase your bet size after a string of wins. That is the moment, when you should stop playing immediately and certainly not increase your bet size. So the approach is particularly harmful, if consecutive bets are negatively correlated, such as is the case with Black Jack.

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Question No.1 : Are price moves at a given resolution (1 sec, 1 min , 30 min, etc.) positively or negatively correlated? Or otherwise put, is the market trending or range bound?

Question No. 2: Are your trades positively correlated, negatively correlated or not correlated?

If you can answer these questions, you are a certainly a genius!

However, what could be done, is to evaluate your trade history and check for auto-correlation. But you would need a rather large sample, as correlations are typically changing a lot.

So you should assume that your trades are not correlated, or that the current correlation cannot be established. In this case the concept of progressive betting does not make sense.

This is the whole point: Progressive betting is the key to success in playing Black Jack, but if you apply it to trading, well, you should better read the book by Ralph Vince to focus on money management and forget about progressive betting.

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To push the thought experiment a bit further, let's assume you have a system with an overall positive expectancy of $50 per trade and an overall success rate of 60%.

If you break the data down further, however, let's say you can classify this as two different systems. Sometimes the system is actually a loser, with an expectancy of ($100) per trade and a success rate of 50%. During successful periods, the system has an expectancy of $200 per trade and a success rate of 70%. The difference is whether market conditions favor the system's approach on a given day.

The key question for the trader is whether, right now on any given day, you're trading the losing system or the winning system. One thing that is clear is that you would expect there to be more (and longer) strings of winners under the winning system than under the losing system. In fact, a sizeable cluster of wins over any given period would be the primary evidence you'd use to determine which system had been in place over that period. If wins are occurring more frequently (or perhaps more accurately, if expectancy is higher), this is what tells you that conditions were favorable for your trading approach over that period.

Might it therefore make sense to increase bet size when you've noticed more winners, or higher expectancy, across the most recent sample of trades?

I don't expect a definitive answer, just raising this as a thought to mull over. I think the answer really depends on your view of how frequently market conditions change. If conditions can be expected to change constantly, so it's basically random whether or not the system will be favored on a given day, then that would be one thing. If conditions can be expected to linger for weeks or months at a time, that would be another. These are empirical questions that you can't really answer with a thought experiment, but that may be worth thinking about.

Hi guys, I'm new here and apologize if this has already been posted as I didn't read this entire thread, but Yale has an entire semester on game theory at academic earth. Lot's of math but good stuff.

I'd post the link but this is my first post.

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Unless I've missed something, I think the issue isn't that the formula is flawed per se, but rather that the results are not optimal.

This is because the profits posted in those charts would be higher if you used a larger bet on all trades than if you used a larger bet only during strings of wins. To see this clearly you'd need another chart showing the results of a fixed position size using a larger bet.

Increasing bet size after a win would make sense if the odds of a win increased with each win, but the assumption in these examples is that the odds of a win are fixed. In that case what you want to do is find the optimal bet size and use it on each trade since each trade is equally likely to give you a profit.

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You are talking about a winning and a losing system. If the both the winning system and the losing system persists over a longer period of time, this simply means

- that you have streaks of bets with a higher expectancy, and streaks of bets with a lower expectancy
- or otherwise put that the trades a positively correlated

Now assuming that the bets of the winning and the losing system taken my themselves do not show any correlation, the overall system will show positively correlated bets. This can be exploited by adapting position sizing via progressive betting.

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I have not checked them, but I think that @Nickemp's results are genuine. So what he clearly showed is that his progressive betting approach increased the returns relative to the win rates.

The point is not the (correct) result, but the explanation.

By using the progressive betting system, he also doubled position size. For any system with a positive expectancy, if you double your position size, you will double your returns. If you trade 1 contract over a year and get a return of $ 30,000, you will possibly earn $ 60,000 when trading 2 contracts. This also implies higher risk.

When you apply a progressive betting system to uncorrelated bets (which makes no sense), you may still benefit from the side effect of increased position size. But then you do not need the system, you can simply double your position size and leave it that way, which would even reduce the dispersion of the cloud shown on the graph presented by Nickemp.

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I thought the strings would equal out because they are based on a 50% chance. It doesn't intuitively make sense to me either and you explained the reasons why much better than I could have. Most of my "thanks" have been to you btw for taking the time to be helpful so often.

Here's where I am now.. If the expectancy of extended strings to occur regularly is valid and the loser at the end doesn't wipe out the profit margin - increasing the position size accordingly is no longer an uncorrelated bet.

I bet others will eventually tinker with and replicate Nickemp's work and that might be more convincing.

Maybe it's just over my head - I can't calculate what the results would be for a larger fixed position size on all trades in my head - but it doesn't make sense to me.

If raising the wager 50% at the start of a string is the optimal betting size; the success of isn't because you have any better chance on that next flip. Rather the combination of strings occurring naturally, compounded with the continual increase in leverage, would make the average return greater on whatever the likely hood of having long enough strings or/and enough of them.

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If you look at nickemp's chart in post # 75, the red dots represent the expected return using a fixed bet of $500.

For instance, the chart shows that the expected return is about $75,000 if you have a 55% success rate and bet $500 each time. If you were to double that bet to $1000 rather than $500, the expected return would double to $150,000. The distribution of blue dots suggests that the string betting system with a $500 starting bet might give you a $150,000 return, or it might not, depending on how the strings play out.

So here's the question: assuming you want to target that $150,000 return, what's the best way to do it:

1. Use a default $500 bet size, and increase it after each win, so that some bets are $500 and others are $4000 or more depending on the outcome of the preceding bets; or

2. Use a default $1000 bet size and don't alter it based on the outcome of the preceding bets.

I would expect choice 2 to give you a more predictable and stable equity curve, without those gut-wrenching large losses that occur at the end of each string of wins under the progressive system. So if you can afford to make your default bet $1000, in most cases I'd see this as preferable to the string system. Unless, that is, the results of your trades are correlated, which is a question I'm personally still grappling with.

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We have now discussed it over and over. Let me just state again the essential point to understand progressive betting.

Essentials

(1) A progressive betting system uses the outcome of the prior trade to establish the position size of the following trade. This is the definition of progressive betting.

(2) If the outcome of the following trade does not depend on the prior trade(s), this does not make sense.

(3) If there is a dependency, progressive betting can have a positive or negative impact.

Side Effect

A progressive betting system always has an impact on position size. If you increase position size, you increase the returns for profitable bets, but increase the losses for non-profitable bets. Your returns will be better but your risk of ruin will be higher as well. The side effect can also be observed for non-correlated bets.

Martingale and Anti-Martingale systems

A Martingale system increases the bet size after a loss. It can be used when the outcomes of consecutive bets are negatively correlated. Example: Black Jack.

An Anti-Martingale system increases the bet size after a win. It can be used when the outcome of consecutive bets are positively correlated. Example: Pyramiding Trades

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This is a crucial point. It really depends on your trading system and whether it is the type of trading system where the result of one trade is influenced by the previous trade or not.

For example, a few types of systems that usually will have dependencies include: always-in systems, trend-following systems that take every trade without skipping any (winners are more likely after a series of losers and vice versa), and mean reversion systems that take repeated entries on the same MR setup even if the previous entries have failed (each consecutive attempt to catch the same falling knife will have higher probability of success, and if you keep trying, eventually you will catch that falling knife)

If, however, your trading system is the type that is not usually in the market and waits for a certain setup to occur, then the chances are smaller that there will be a dependency between the trades.

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So far, a few brokers have offered CL in their portfolio.

It is expensive. About $30 USD per lot transacted.

I have been using a Mini-Lot broker, 10,000 units = 1 lot for my automated GBPUSD trading. So far it is good because I have a Grid/Martingale …

This sounds very impressive, but is the simplest of all progressive betting systems. You will know this from Roulette:

You start wit $ 1 and put it on black. If you win you get another $1. If you lose, you double up and bet $2. You continue doubling up until you win. This way you will always win. For example if you experience 10 consecutive losses (probability is around 1:784) and then win, your profit is Profit = $ - 1 - 2 - 4 - 8 - 16 - 32 -64 - 128 - 256 - 512 + 1024 = $ 1.

The problem of such bets is the asymmetric risk profile. Every casino has a limit, and let us assume that the limit of the casino is $ 50,000. With 16 consecutive losses (probability around 1: 42,777) you cannot continue to play. Statistically you will have 42,776 winning strings with a profit of 1$ and one losing string, which comes at a combined cost of $ 65.535. The difference is the edge of the casino.

Now, if you want to apply that wonderful system to trading, you can get yourself a Grid Martingale Expert Advisor. It has different levels for averaging down, so you can play it in the way you can play Roulette. Now, if the bets are uncorrelated, your account will prosper for some time, before experiencing a sudden death.

The only way to make money with a Martingale System would be to find a system that produces bets, which are negatively correlated. Does anybody have some evidence that such a negative correlation can be exploited?

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I spent weeks studying Vince 8 years ago and setting up excell files of a 35+ year trading history to calculate optimum f on my personal trading history.

Well worth the effort, and it uncovered that my bet sizes were often way too big. It also reduced my overall market exposure, and confirmed a kind of anti-Martingale approach of the type Vince discusses in his "Leveraged Trading Space"

But even @ optimum f, I still didn't like the drawdowns, and the potential drawdowns, especially when trading in highly correlated assets in leveraged positions, although the run ups were exciting.

Then I found that adjusting optimum f for the biggest drawdown /size of loss I'm willing to suffer helped both my trading and my disposition.

Thus, I now use Optimum f % x total equity bet size/largest loss % =bet size/1% of equity. Such sizing also permitted more reliance on trading signals for exits rather than stops.

Then I increased the number of systems I traded and expanded the asset classes involved in an effort to reduce the overall correlation of the different bets in the portfolio.

Seems to work OK and add consistency, but there is more effort to time and monitor many more asset classes/positions, plus be worried about how everything seems to be correlated these days. A lot of work for a lone trader.

Indeed, a resource on correlation coefficients would be most helpful.

Geo

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