How advanced mathematics and gaming theory can help you as a trader

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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).

The following 3 users say Thank You to fluxsmith for this post:

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?

The following 2 users say Thank You to worldwary for this post:

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.

The following 13 users say Thank You to Fat Tails for this post:

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.

The following user says Thank You to RM99 for this post:

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.

The following user says Thank You to RM99 for this post:

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.

The following 2 users say Thank You to Fat Tails for this post:

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.